Practice Problems on the Linear Momentum Equations
COLM_01
A frequently used hydraulic brake consists of a movable ram that displaces water from a slightly larger cylinder, as
shown in the figure. The cross-sectional area of the cylinder is Ac and the cross-sectional area of the ram is Ar. The
ram velocity, V, remains constant. Assume that the gap between the cylinder and the ram is much smaller than the
displacement of the ram x.
a.
b.
Determine the water velocity as it leaves the cylinder.
Determine the force F on the ram in terms of Ar, Ac, and V. Assume that the cylinder is initially full of water,
that gravitational effects are negligible, and that the water exits the cylinder to atmospheric pressure.
Ac
water filled
cylinder
F
V
Ar
x
Answer(s):
1
A

Vout
Ar

  c  1
V
Ac  Ar  Ar

Ac Ar
F  V 2  32 Ar  Ac 
2
 Ac  Ar 
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Practice Problems on the Linear Momentum Equations
COLM_02
A sonar buoy is being tested in a wind tunnel with a circular test section as shown in the figure. The air in the tunnel
has a density of 1.2 kg/m3 and the test section radius is R = 0.5 m. Measurements at location A indicate that the
incoming velocity is uniform with a velocity of U = 10 m/s while the downstream velocity, measured at location B,
is zero at the centerline and increases linearly with radius, r, measured from the tunnel’s centerline. A U-tube
manometer filled with water (with a density of 1000 kg/m3) is used to measure the pressure difference between
sections A and B (the pressure is assumed uniform at each section). The elevation difference between the two legs
of the manometer is H=0.03 m as shown in the figure.
a.
b.
Determine the maximum velocity, Umax, at section B.
Determine the drag acting on the buoy.
Section A
Section B
R
r
U
H
water
Answer(s):
U max  32 U 
 Fdrag   H2 O gH  R 2  8 airU 2 R 2
 Fdrag =
220 N
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Practice Problems on the Linear Momentum Equations
COLM_03
An incompressible, viscous fluid with density, , flows past a solid flat plate which has a depth, b, into the page.
The flow initially has a uniform velocity U, before contacting the plate. After contact with the plate at a distance x
downstream from the leading edge, the flow velocity profile is altered due to the no-slip condition. The velocity
profile at location x is estimated to have a parabolic shape, u=U((2y/)-(y/)2), for y   and u=U for y where 
is termed the “boundary layer thickness.”
u=U, y
streamline

y
h
plate has depth, b,
into the page
u=U
x
1.
2.
u=U((2y/)-(y/2), y
Determine the upstream height from the plate, h, of a streamline which has a height, , at the downstream
distance x. Express your answer in terms of .
Determine the force the plate exerts on the fluid over the distance x. Express your answer in terms of , U, b,
and . You may assume that the pressure everywhere is p. The force the drag exerts on the plate is called the
“skin friction” drag.
Answer(s):
h  23 
F  152 U 2 b
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Practice Problems on the Linear Momentum Equations
COLM_04
Wake surveys are made in the two-dimensional wake behind a cylindrical body which is externally supported in a
uniform stream of incompressible fluid approaching the cylinder with velocity, U.
U
y
x
wake
b(x)
u = U - A(x) cos[y/b(x)]
U
U
The surveys are made at x locations sufficiently far downstream of the body so that the pressure across the wake is
the same as the ambient pressure in the fluid far from the body. The surveys indicate that, to a first approximation,
the velocity in the wake varies with lateral position, y, according to:

A x
u
y 
1
y
1
cos 
 1

 where  
U
U
b
x
2
b
x
2






The quantities A(x) and b(x) are the centerline velocity defect and wake width, respectively, both of which vary with
position, x. If the drag on the body per unit distance normal to the plane of the sketch is denoted by D and the
density of the fluid by , find the relation for b(x) in terms of A(x), U, , and D.
Answer(s):
b  x 
D
A x 
 2

U 2 A  x  
2 
  U 2U 
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Practice Problems on the Linear Momentum Equations
COLM_05
A hydraulic jump is a sudden increase in the depth of a liquid stream (which in this case we assume is flowing over
a horizontal stream bed with atmospheric pressure air everywhere above the liquid):
g
free surface
h2
h1
U2
U1
The depth increases suddenly from h1 to h2 downstream of the jump. The jump itself is often turbulent and involves
viscous losses so that the total pressure downstream is less than that of the upstream flow.
a. Find the ratio of the depths, h2/h1, in terms of the upstream velocity, U1, the depth, h1, and g, the acceleration
due to gravity. Assume the flows upstream and downstream have uniform velocity parallel to the stream bed
and that the shear stress between the liquid and the stream bed is zero. The liquid is incompressible.
b. What inequality on the value of U12/(gh1) must hold for a hydraulic jump like this to occur?
Answer(s):

h2
1
1 2U12
 

h1
2
4 gh1
U12
1
gh1
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Practice Problems on the Linear Momentum Equations
COLM_06
A turbojet engine in a wind tunnel receives air at a velocity of Ui = 100 m/s and a density of i = 1 kg/m3. The
velocity is uniform and the cross-sectional area of the approaching stream which enters the engine is Ai = 0.1 m2.
The velocity of the exhaust jet from the engine, however, is not uniform but has a velocity which varies over the
cross-section according to:
uo = 2U (1-r2/R2)
where the constant U = 600 m/s and R is the radius of the jet cross-section. Radial position with the axi-symmetric
jet is denoted by r. The density of the exhaust jet is uniformly 0 = 0.5 kg/m3.
a.
b.
c.
Determine the average velocity of the exhaust jet.
Find the thrust of the turbojet engine.
Find what the thrust would be if the exhaust jet had a uniform velocity, U.
Assume the pressures in both the inlet and exhaust jets are the same as the surrounding air and that mass is
conserved in the flow through the engine (roughly true in practice).
r
velocity, Ui,
incoming area, Ai,
density, i
R
velocity, uo = 2U(1-r2/R2)
density, o
mounting stand
Answer(s):
 uo  U
 T  iU i2 Ai  43 U  1 ; T = 7000 N
Ui 

T  iU i2 Ai  U  1 ; T = 5000 N
 Ui 
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Practice Problems on the Linear Momentum Equations
COLM_07
A jet of water is deflected by a vane mounted on a cart. The water jet has an area, A, everywhere and is turned an
angle  with respect to the horizontal. The pressure everywhere within the jet is atmospheric. The incoming jet
velocity with respect to the ground (axes XY) is Vjet. The cart has mass M. Determine:
a. the force components, Fx and Fy, required to hold the cart stationary,
b. the horizontal force component, Fx, if the cart moves to the right at the constant velocity, Vcart (Vcart<Vjet)
c. the horizontal acceleration of the cart at the instant when the cart moves with velocity Vcart (Vcart<Vjet) if no
horizontal forces are applied

A
Vjet
Vcart
Y
Fx
X
Fy
Answer(s):
2
2
Fx  Vjet
A 1  cos   ; Fy  Vjet
A sin   Mg

Fx   V jet  Vcart
2
 A 1  cos  
 Vjet  Vcart  A 1  cos  
2
a
M
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Practice Problems on the Linear Momentum Equations
COLM_09
A fluid enters a horizontal, circular cross-sectioned, sudden contraction nozzle. At section 1, which has diameter
D1, the velocity is uniformly distributed and equal to V1. The gage pressure at 1 is p1. The fluid exits into the
atmosphere at section 2, with diameter D2. Determine the force in the bolts required to hold the contraction in place.
Neglect gravitational effects and assume that the fluid is inviscid.
bolts
D2
V1
D1
p1
atmosphere
Answer(s):
Fbolts  V12
2
 D12  D1 

 D12

  1  p1,gage
4  D2 
4



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Practice Problems on the Linear Momentum Equations
COLM_10
Water is sprayed radially outward through 180 as shown in the figure. The jet sheet is in the horizontal plane and
has thickness, H. If the jet volumetric flow rate is Q, determine the resultant horizontal anchoring force required to
hold the nozzle stationary.
Q
H
R
Answer(s):
2
 Q 
Fx  2  
 RH
  RH 
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Practice Problems on the Linear Momentum Equations
COLM_11
A variable mesh screen produces a linear and axi-symmetric velocity profile as shown in the figure. The static
pressure upstream and downstream of the screen are p1 and p2 respectively (and are uniformly distributed). If the
flow upstream of the mesh is uniformly distributed and equal to V1, determine the force the mesh screen exerts on
the fluid. Assume that the pipe wall does not exert any force on the fluid.
Section 1
variable mesh
screen
Section 2
p1
2R
p2
V1
Answer(s):
F   8 V12 R 2   p1  p2   R 2
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Practice Problems on the Linear Momentum Equations
COLM_12
The tank shown rolls along a level track. Water received from a jet is retained in the tank. The tank is to accelerate
from rest toward the right with constant acceleration, a. Neglect wind and rolling resistance. Find an algebraic
expression for the force (as a function of time) required to maintain the tank acceleration at constant a.
initial mass of cart and water,
M0
A
V
F
U
Answer(s):


2
F   V  at  A  a  M 0   Vt  12 at 2 A


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Practice Problems on the Linear Momentum Equations
COLM_14
A cart with frictionless wheels holds a water tank, motor, pump, and nozzle. The cart is on horizontal ground and
initially still. At time zero the cart has a mass M0 and the pump is started to produce a jet of water with constant
area Aj, velocity Vj at an angle  with respect to the horizontal. Find and solve the equations governing the mass and
velocity of the cart as a function of time.
Answer(s):
M CV  M 0  V j A j t
 V j Aj t 
U  V j cos  ln  1 

M0 

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Practice Problems on the Linear Momentum Equations
COLM_15
An incompressible, inviscid liquid flows steadily through a circular pipe of constant diameter, D. The pipe contains
a section with porous wall of length, L. There, liquid is removed at constant radial velocity, v0, with no axial
component of velocity. The liquid speed and pressure at the entrance to the porous section are U0 in the axial
direction and p0, respectively. Assume v0 << U0 so the flow is uniform at each cross-section.
a.
b.
c.
Obtain an algebraic expression for the distribution of axial velocity, U(x), for 0  x  L.
Obtain an algebraic expression for the distribution of pressure, p(x), for 0  x  L.
Sketch the velocity and pressure distributions as a function of distance, x, for 0  x  L.
L
x
D
U0, p0
v0
Answer(s):
 v  L  x 
U
 1 4 0   
U0
 U0   D   L 
2
 v   L  x 
 v   L x
p  p0
 16  0      32  0     
2
1

U
U
D
L
0
 0    
 U0   D   L 
2
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2
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Practice Problems on the Linear Momentum Equations
COLM_17
A rocket cart that is initially at rest and has an initial mass of M0 is to be fired and is to have a constant acceleration
of a. To accomplish this, the exhaust gases will be deflected through an angle  which varies as a function of time.
 and a
The rocket exhausts gases with constant density through a constant area nozzle at a mass flow rate of m
constant speed, Ue, relative to the rocket.
a.
b.
c.
Find an expression for cos() as a function of time t.
Determine the time, tmin, at which cos() is a minimum.
Find max.
Ue

Answer(s):
 a
 M 0  mt
cos  
 e
mU
tmin 
M0
m
cos  max  0   max 

2
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Practice Problems on the Linear Momentum Equations
COLM_18
In an attempt to model the speed of a tsunami wave in the deep ocean, consider the propagation of a small
amplitude, solitary wave front moving with speed, c, from right to left as shown in the figure below. Neglect the
effects of surface tension. The liquid is initially at rest but after the wave passes by, the fluid behind the wave has a
small velocity, dV, in the same direction as the wave.
Derive an expression for the wave speed, c. You may neglect the shear forces the channel bed and the atmosphere
exert on the liquid. Hint: Consider choosing a steady frame of reference when analyzing the problem.
c
dV
g
h
Answer(s):
 c  gh
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Practice Problems on the Linear Momentum Equations
COLM_19
A flat plate of mass, M, is located between two equal and opposite jets of liquid as shown in the figure. At time t=0,
the plate is set into motion. Its initial speed is U0 to the right; subsequently its speed is a function of time, U(t). The
motion is without friction and parallel to the jet axes. The mass of liquid that adheres to the plate is negligible
compared to M.
Obtain algebraic expressions (as functions of time for t>0) for:
a.
the velocity of the plate and
b.
the acceleration of the plate.
c.
What is the maximum displacement of the plate from its original position?
Express all of your answers in terms of (a subset of) U0, V, A, , M, and t.
A
,V
A
,V
U(t)
plate with mass, M
Answer(s):
U
 4 VAt 

 exp  

U0
M 

a
4 U 0VA
dU
 4 VAt 

exp  

dt
M
M 

 xmax 
MU 0
4 VA
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Practice Problems on the Linear Momentum Equations
COLM_22
A spherical droplet of liquid, with density, d, moves horizontally (no gravity) through a stationary vapor cloud. As
it moves, additional liquid accumulates on the upstream side of the droplet at a mass rate, m , given by:
m  d R 2V
where  is a dimensionless constant, R is the droplet radius, and V is the droplet velocity. In addition, there is a drag
force acting on the droplet given by:
D  cd 1 2  vV 2 R 2
where cd is the dimensionless drag coefficient (assumed to be a constant) and v is the surrounding vapor cloud
density (also assumed constant).
a.
b.
Determine the time rate of change of the droplet radius as a function of  and V.
Determine (but do not solve) the differential equation describing the time rate change of the drop radius in
terms of cd, , v, and d. Do not include V in this differential equation.
Answer(s):
dR 

V
dt 4
  cd v
 1  dR  2


3
1

 
 0
2
dt 2
  d
 R  dt 
d 2R
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Practice Problems on the Linear Momentum Equations
COLM_23
Two parallel plates of width, 2a, are separated by a gap of height, h. The upper plate approaches the lower plate at a
constant speed, V. The space between the plates is filled with a frictionless, incompressible gas of density,.
Assume that the velocity is uniform across the gap width (y direction) so that u=u(x, t).
Obtain algebraic expressions for:
a.
the velocity distribution, u(x, t).
b.
the pressure distribution in the gap, p(x, t). The pressure outside of the gap is atmospheric pressure. Note:
You do not need to use Bernoulli’s equation to solve this problem.
y
patm
h
V
u(x, t)
patm
x
a
a
Answer(s):
x
u  V
h
 a  2  x  2 
p  patm

2
     
2
1
2 V
 h   h  
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Practice Problems on the Linear Momentum Equations
COLM_24
A weir discharges into a channel of constant breadth as shown in the figure. It is observed that a region of still water
backs up behind the jet to a height a. The velocity and height of the flow in the channel are given as V and h,
respectively, and the density of the water is . You may assume that friction and the horizontal momentum of the
fluid falling over the weir are negligible.
a
h
V
What is the height a in terms of the other parameters?
Answer(s):
a
  1  2Fr 2 where Fr = V/(gh)1/2
h
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Practice Problems on the Linear Momentum Equations
COLM_25
A tank with a reentrant orifice called a Borda mouthpiece is shown in the figure. The reentrant orifice essentially
eliminates flow along the tank walls, so the pressure there is nearly hydrostatic. Calculate the contraction
coefficient, Cc=Aj/A0. You may reasonably assume that the fluid is inviscid and incompressible.
free surface
Aj
A0
Answer(s):
Aj 1
 Cc 

A0 2
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Practice Problems on the Linear Momentum Equations
COLM_26
A wedge with a vertex angle of 2 is inserted into a jet of water of width, b, density, , and velocity, U, as shown in
the sketch. The angle of attack, , of the wedge is also defined in the sketch. After impinging on the wedge, the
single incident jet is divided into two jets, both of which leave the back edges of the wedge with velocity, U. The
widths of the two departing jets are b and (1-)b as indicated in the figure. It is assumed that the flow is planar,
gravity may be neglected, and the pressure in the surrounding air is everywhere atmospheric.
a.
b.
c.
Find the lift and drag on the wedge per unit length normal to the sketch as functions of , U, b, , , and .
Note that drag and lift are defined as the forces on the wedge that are, respectively, parallel and perpendicular
to the direction of the incident jet.
While holding the other parameters fixed, find the angle of attack,  at which the lift is zero.
If the wedge is moved in a direction perpendicular to the incident jet while, U, b, , and  remain fixed then
 will vary. There is one such position at which the lift is zero; what is the value of  at this position in terms
of  and ? If the wedge were free to move in such a way, would this position represent a position of stable
or unstable equilibrium?
U
, U
b

b


b
U
Answer(s):
 D  U 2b 1   cos      1    cos     
 L  U 2b 1    sin       sin     
   tan 1  2  1 tan  
1  tan  
   1 

2
tan  
’ is an unstable equilibrium point
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Practice Problems on the Linear Momentum Equations
COLM_27
Gravel is dumped from a hopper, at a rate of 650 kg/s, onto a moving belt. The gravel then passes off the end of the
belt as shown in the figure. The drive wheels are 80 cm in diameter and rotate clockwise at 150 rpm. Neglecting
system friction and air drag, estimate the power required to drive this belt.
Answer(s):
2
W  FV  m  1  D 
2
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Practice Problems on the Linear Momentum Equations
COLM_28
A block of mass, M=10 kg, with rectangular cross-section is arranged to slide with negligible friction along a
horizontal plane. As shown in the sketch, the block is fastened to a spring that has stiffness such that F=kx where
k=500 N/m. The block is initially stationary. At time, t=0, a liquid jet begins to impinge on the block (the jet
properties are also shown in the sketch). For t>0, the block moves laterally with speed, U(t).
a. Obtain a differential equation valid for t>0 that could be solved for U(t) and X(t). Do not solve.
b. State appropriate boundary conditions for the differential equation of part (a).
c. Evaluate the final displacement of the block.
=1000 kg/m3
V=30 m/s
A=100 mm2
k
M
X
Answer(s):
d2X A
dX 
k
X 0

V 
 
2
M 
dt  M
dt
A 2
Xf 
V
k
2
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Practice Problems on the Linear Momentum Equations
COLM_32
A cart hangs from a wire as shown in the figure below. Attached to the cart is a scoop of width W (into the page)
which is submerged into the water a depth, h, from the free surface. The scoop is used to fill the cart tank with water
of density, .
cart has initial mass, M0
and initial velocity, V0
V
scoop has width, W, into the page
stagnant water
h
a.
b.
Show that at any instant V=V0M0/M where M is the mass of the cart and the fluid within the cart.
Determine the velocity, V, as a function of time.
Answer(s):
M
V  V0 0
M
V
1

V0
2  hWV0 t
M0
1
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Practice Problems on the Linear Momentum Equations
COLM_34
Two water jets strike each other and merge into a single jet as shown in the figure. Determine the speed, V, and
direction, , of the resulting combined jet. Gravity is negligible.
V

V2
d2
90
d1
V1 = 10 ft/s
V2 = 15 ft/s
d1 = 0.1 ft
d2 = 0.05 ft
V1
Answer(s):
V 2d 2
tan   12 12
V2 d 2
V3 
V d
2
1 1
V22 d 22
 V2 d 22  cos 
C. Wassgren, Purdue University
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Last Updated: 2010 Sep 16
Practice Problems on the Linear Momentum Equations
COLM_35
Police attempt to control demonstrators by using a “water cannon” mounted on a moving truck. The cannon shoots
a steady stream of water, with a diameter of 25 mm and speed of 10 m/s (relative to the truck). The truck moves at
speed of 3 m/s toward the demonstrators. Calculate the maximum force that could be exerted on a demonstrator by
the water stream assuming that the water jet is not deflected back toward the water cannon and the jet hits the
demonstrator perpendicularly. Describe the manner in which the force would vary if the stream were to impinge at
an angle other than perpendicular.
Answer(s):

F   V jet  Vtruck


2
2
 D jet
4

F   V jet cos   Vtruck V jet  Vtruck cos 
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
2
 D jet
4
Page 26 of 49
Last Updated: 2010 Sep 16
Practice Problems on the Linear Momentum Equations
COLM_36
Liquid falls steadily and vertically into a short horizontal rectangular open channel of width b. The total volume
flow rate, Q, is distributed uniformly over area bL. Neglect viscous effects.
a.
Obtain an expression for h1 in terms of h2, Q, and b.
b.
Plot the dimensionless surface profile, h/h1, as a function of the dimensionless position, x/L, for a variety of
parameters.
y
Q
h2
h1
Q
x
L
Answer(s):
 h1  h22 
2Q 2
gh2 b 2
dh
2

3

dx  x   

2  h 
   1  Fr  2  
h
x
 


where x’ = x/L, h’ = h/h1, and Fr2 = gb2h13/Q2 (this is a Froude number).

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Practice Problems on the Linear Momentum Equations
COLM_39
A stream of incompressible liquid moving at low speed leaves a nozzle pointed directly upward. Assume the speed
at any cross section is uniform and neglect viscous effects. The speed and area of the jet at the nozzle exit are V0
and A0, respectively. Apply conservation of mass and the momentum equation to a differential control volume of
length dz in the flow direction. Derive expressions for the variations of jet speed and area as functions of z.
Evaluate the vertical distance required to reduce the jet speed to zero. (Take the origin of coordinates at the nozzle
exit.)
Answer(s):
V  V02  2 gz ; zV=0 = V02/(2g)
A
V0 A0
V02  2 gz
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Practice Problems on the Linear Momentum Equations
COLM_40
Incompressible fluid of negligible viscosity is pumped, at total volume flow rate Q, through a porous surface into the
small gap between closely spaced parallel plates as shown. The fluid has only horizontal motion in the gap.
Assume uniform flow across any vertical section. Obtain an expression for the pressure variation as a function of x.
L
x
h
V(x)
Assume a depth w
into the page.
Q
Answer(s):
p  patm
 
 Q hw
2
1 x
  
4 L
2
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Practice Problems on the Linear Momentum Equations
COLM_42
Flying Elvi, 20 of them in all, jump out of an airplane at a rate of one Elvis, weighing 255 lbm, every 5 seconds. If
the airplane is flying horizontally at a velocity of 120 mph and tries to accelerate at a rate of 1 ft/s2, determine the
change in the thrust that must be supplied by the airplane propellers as a function of time until all of the Elvi have
left the building plane.
Assume that there is a drag force acting on the plane that can be modeled by FD = -kV2 where k = 0.2 lbf/(ft2/s2) and
V is the velocity of the plane relative to the air. The air density at an altitude of 2.5 miles is approximately 0.61
times the air density at sea level. You may assume that the mass rate at which fuel is burned is very small in
comparison to the mass rate at which Elvi leave the plane. The plane weight at altitude not counting Elvi is 30000
lbm.
120 mph
altitude = 2.5 miles
Elvis weight = 255 lbm
Answer(s):
 
T   M 0  mt
T
=
dV
dt
1090 lbf – (1.6 lbf/s) t (0  t  100 s since there are 20 Elvi)
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Practice Problems on the Linear Momentum Equations
COLM_43
A model solid propellant rocket has a mass of 69.6 gm, of which 12.5 gm is fuel. The rocket produces 1.3 lbf of
thrust for a duration of 1.7 sec. For these conditions, calculate the maximum speed and height attainable in the
absence of air resistance. Plot the rocket speed and the distance traveled as functions of time.
Answer(s):
Umax = U(t = t’ = 1.7 sec) = 139.2 m/s
hmax = h(t = tm = 15.9 sec) = 1100 m
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(h(t = t’) = 114 m)
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Practice Problems on the Linear Momentum Equations
COLM_46
A jet of water sprays into a container as shown in the figure. The water jet is deflected as it enters the container so
that all of the water that enters the container remains in the container. The initial mass of the container and the water
within it is M0. The velocity of the container is given by U.
a.
Determine the velocity, V2, of the fluid just as it enters the container in terms of V1, g, and h.
For the remaining questions, express your answers in terms of V1, V2, U, A1, , g, and h.
b.
c.
d.
e.
Determine the mass flow rate of the water entering the container.
Determine the equation for the mass, M, of the container plus the water within the container at time t. You do
not need to solve any integrals you might have.
Write out (but do not solve) the differential equation describing the container acceleration, dU/dt. You do not
need to substitute your result from part (c) into this differential equation, i.e. leave terms involving mass in
terms of M.
Solve (numerically) for the container height, h, as a function of time, t, for the following conditions:
M0 = 1 kg
V1 = 10 m/s
A1 = 4.0*10-4 m2
 = 1000 kg/m3
g = 9.81 m/s2
h0 = 1 m
U0 = 1 m/s
U
g
V2
h
V1
A1

Answer(s):
V2  V12  2 gh
m in  

t t


V1
 ;  M CV  M 0   
V12  2 gh  U A1 
 V 2  2 gh 
t 0
 1





V1
 dt
V12  2 gh  U A1 
 V 2  2 gh 
 1


dU  V2  U  A2

g
dt
M CV
2
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Practice Problems on the Linear Momentum Equations
COLM_48
a.
Construct from first principles an equation for conservation of mass governing the planar flow (in the xy plane)
of an incompressible liquid lying on a flat horizontal plane. The depth, h(x,t), is a function of position, x, and
time, t. Assume that the velocity of the fluid in the positive x-direction, u(x,t), is independent of y.
b. Now apply conservation of linear momentum in the x-direction using the same control volume. Assume that the
ground does not exert a shear stress on the fluid.
The resulting partial differential equations are part of what is known as Shallow Water Wave Theory.
g
free surface
liquid
h(x,t)
y
x
u(x,t)
Answer(s):
h   uh 

0
t
x
  uh 
t

    gh h
 u2h
x
x
or
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u
u
h
u
 g
t
x
x
Page 33 of 49
Last Updated: 2010 Sep 16
Practice Problems on the Linear Momentum Equations
COLM_51
A cart travels at velocity, U, toward a liquid jet that has a velocity, V, relative to the ground, a density, , and a
constant area, A. The mass of the cart and its contents at time t = 0 is M0 and the cart’s initial velocity is U0 toward
the jet. The resistance between the cart’s wheels and the surface is negligible.
V, A, 

g
U
Determine the mass flow rate into the cart in terms of (a subset of) , A, V, U, g, and .
Determine the acceleration of the cart, dU/dt, in terms of , A, V, U, g, , and M(t) where M(t) is the mass of
the cart and water at time t. You needn’t solve any integrals or differential equations that appear in your
answer.
a.
b.
Answer(s):
m into  
cart
 u
rel
 dA   U  V  A
CS
 U  V  A
dU

dt
M
2
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Practice Problems on the Linear Momentum Equations
COLM_52
A rocket powered sled travels with velocity, U (with respect to the ground), up an incline that is at an angle,  with
respect to the horizontal. The rocket exhaust, directed in the horizontal direction, has a constant velocity, V (with
respect to the rocket), and a mass flow rate, m , that varies with time, t, according to:
 
t 
t T
 m 1 
m   0  T 

0
t T

where m 0 is a constant and T is the time at which all of the fuel has been expended.
U
X
g

V , m

Assuming that the initial mass of the rocket sled and fuel is M0 and the initial rocket velocity up the slope is zero,
determine:
a.
b.
c.
d.
the mass of the rocket sled as a function of time,
the cart acceleration in the X direction (shown in the figure) for all times, and
the cart velocity in the X direction for all times (you needn’t solve any integrals that occur),
Determine the time (t  T) at which the cart velocity will be zero.
You may neglect aerodynamic drag and rolling resistance.
Answer(s):

 1 t2 
 M 0  m 0  t 
 t T
M t   
 2T 
 M  1 m T
t T
0
0
2




m 1  t V cos 
T
 g sin   0
dU 
 1 t2 

M 0  m 0  t 

dt 
 2T 

 g sin 

t T
t T

 m 0 t  1 t  
  g sin   t  V cos  ln 1 
1 
 t  T
U t   
 M 0  2 T 

U T   g sin   t  T 
t T

tU  0  T 
U T 
g sin 
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Practice Problems on the Linear Momentum Equations
COLM_54
a.
b.
c.
Using an integral approach, write the differential equation governing the motion of an inviscid,
incompressible fluid (with density ) oscillating within the U-tube manometer shown. The manometer crosssectional area is A.
What is the natural frequency of the fluid motion?
What are the implications of this result for making time-varying pressure measurements using a manometer?
tube ends are open to the
atmosphere
g
h1
h2
incompressible, inviscid fluid
with density 
Assume this distance is negligible
compared to h1+h2.
Answer(s):
d 2 z 2g

z0
L
dt 2
z V
 2g 
 2g 
L
sin  t
  z0 cos  t
; 
2g
L 
L 


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2g
L
Page 36 of 49
Last Updated: 2010 Sep 16
Practice Problems on the Linear Momentum Equations
COLM_55
The toy boat (shown below) travels at a constant speed, Vboat. The boat is propelled using a compressed air tank that
issues air at a pressure of 1 atm, temperature of 15 C, velocity of 340 m/s, and diameter 5 mm. Although air drag is
negligible, the hull drag due to the water is significant and varies as kVboat2 where k  15 Ns2/m2. Determine the
speed of the boat.
compressed air jet
Vboat
Answer(s):
p jet
Vboat 
RTjet
2
Vjet
2
 Djet
4
k
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Practice Problems on the Linear Momentum Equations
COLM_56
Air leaves a nozzle with a diameter of 15 mm and strikes the center of a vertically oriented circular plate with a
diameter of 50 mm. The force required to move the plate toward the jet at a constant speed of 0.5 m/s is 10 N.
Determine the pressure in the air supply pipe, which has a diameter of 20 mm.
plate velocity = 0.5 m/s
force = 10 N
pipe diameter = 20 mm
exit diameter = 15 mm
plate diameter = 50 mm
pressure = ?
Answer(s):
=
ppipe
120 kPa (abs) = 19.3 kPa (gage)
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Last Updated: 2010 Sep 16
Practice Problems on the Linear Momentum Equations
COLM_57
Consider the four carts shown below. Each of the carts rests on a frictionless surface, is initially stationary, and is
restricted to move only in the horizontal direction. The pressure surrounding the cart is atmospheric and the flow is
steady and incompressible. In what direction will each device move when it is released? You must provide support
for your answers.
2
1
4
3
Answer(s):
---
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Practice Problems on the Linear Momentum Equations
COLM_59
A bucket of water (with radius, R, and initial mass, M0) is filled by the falling water jet shown in the figure. The
water jet falls under the action of gravity from a large tank of depth, s, and exit area, AE. The bucket can slide
vertically within the tube on a thin lubricating layer of fluid with thickness, t, and dynamic viscosity, . The
pressure everywhere is atmospheric.
s
large tank with exit area, AE
viscous fluid with dynamic
viscosity, , and thickness, t
g
h
Z
Vj
bucket with mass, M, radius,
R, and length, L
L
U
Determine the velocity, Vj, of the water jet (relative to the ground) just before it enters the bucket. Assume
that the bucket at this instant in time is located a distance h below the free surface of the large tank. Express
your answer in terms of the elevation, h, and other pertinent parameters.
Determine the mass flow rate of water entering the bucket, m into bucket , if the bucket speed is U relative to the
ground. Express your answer in terms of the bucket velocity, U, the tank exit area, AE, the elevations s and h,
and other pertinent parameters.
Determine the mass of the bucket with water, M, at time t assuming the initial bucket/water mass is M0. You
need not solve any integrals that appear in your derivation. You may leave your answer in terms of m into bucket
and other pertinent parameters.
Determine the total force (body and surface) acting on the bucket with water if the bucket with water has
velocity U and mass M.
Derive an expression for the bucket/water acceleration in terms of the bucket velocity, U, bucket mass, M, jet
velocity, Vj, mass flow rate, m into bucket , and other pertinent parameters.
a.
b.
c.
d.
e.
Answer(s):
V j  2 gh
m into

bucket


2 gh  U AE
s
h
t t
M  M0 
 m
t 0
into
bucket
dt
U
2 RL
t
U
Mg   2 RL  m into V j  U 
dU
t
bucket

dt
M
Ftotal, z  Mg  
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Practice Problems on the Linear Momentum Equations
COLM_60
A railroad freight car, with an empty mass of M0 and a cross-sectional opening area of A, is accidentally released
down an incline of angle  with respect to the horizontal. As luck would have it, a steady rain is falling at a terminal
velocity V (with respect to the ground) and density . The rain enters the car as it rolls downhill. You may neglect
drag forces and friction acting on the car.
V, 
g
A
U

a.
Determine, in vector form, the velocity of the rain relative to the car using a clearly defined frame of
reference.
Determine the opening area vector using the same frame of reference used in part (a).
Determine the mass flow rate of water entering the car.
Determine the mass of the cart and water, M, as a function of time.
Determine the acceleration of the car down the incline in terms of M, g,  A, V, , and U.
b.
c.
d.
e.
Answer(s):
 u rel,xyz  V sin   U  ˆi  V cos  ˆj (using a coordinate system fixed to the cart and oriented so that x points
downhill)
A xyz  Aˆj

 u
rel
 dA   AV cos 
CS
M  t   M 0  VAt cos 
V sin   U  VA cos  
dU
 g sin  
dt
M
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Practice Problems on the Linear Momentum Equations
COLM_61
Consider a viscous, incompressible, Newtonian, laminar liquid film of depth h, density, , and constant dynamic
viscosity, , flowing steadily under the influence of gravity down an inclined surface as shown in the figure below.
Assume that the liquid’s velocity profile, u(y), is fully developed meaning that it is not a function of distance down
the incline (the x direction).
L
y
g
h
u(y)
Assume that the
flow has a depth, W,
into the page.
dy

x
a.
Write a differential equation for the fluid’s velocity gradient, du/dy, using an analysis based on a differential
control volume of thickness, dy, length, L, and depth into the page of W, as indicated in the figure. (Do not
use the Navier-Stokes equations for this problem.)
Write the appropriate boundary conditions for solving this differential equation.
Solve for the velocity profile, u(y).
b.
c.
Answer(s):

d 2u
  g sin 
2
dy

u  y  0  0


du
 y  h  0
dy
u
gh

 Fr

1 h gh
y
y
sin   2  

2 
h
h
 Re
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Practice Problems on the Linear Momentum Equations
COLM_64
A cart of empty mass, M0, planform (i.e. cross-sectional) area, AC, and horizontal speed, U, moves on a thin viscous
layer of liquid with dynamic viscosity, , density, , and layer thickness, h. A horizontal liquid jet with velocity V1
(with respect to the ground), density, 1, and area A1, is directed into the cart. In addition, the cart moves through a
continuous liquid spray, with density, 2, that has a downward vertical speed, V2 (with respect to the ground).
2,V2
g
1, V1
A1
AC
U
h
a.
b.
viscous Newtonian fluid with
constant dynamic viscosity, and
density,
Determine the rate at which the mass inside the cart is changing with time in terms of M0, 1, V1, A1, 2, V2,
AC, U, h, , , g, or a subset of these variables. You need not evaluate any integrals.
Determine the acceleration of the cart in terms of M0, 1, V1, A1, 2, V2, AC, U, h, , , g, or a subset of these
variables. You need not evaluate any integrals.
Answer(s):
dM CV
 1 V1  U  A1   2V2 AC
dt
1 V1  U  A1   2V2UAC  
2
dU

dt
U
AC
h
M CV
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Practice Problems on the Linear Momentum Equations
COLM_65
Two long trains carrying coal are traveling in the same direction side by side on separate tracks. One train is
moving at 40 ft/s (train 1) and the other at 50 ft/s (train 2). In each coal car a man is shoveling coal and pitching it
across to the neighboring train. The rate of coal transfer from train 1 to train 2 is 8000 lbm/min for each 100 ft of
train length, and from train 2 to train 1 is 6000 lbm/min for each 100 ft of train length. Find the extra force that must
be exerted on each train (per unit length) in order to maintain constant train speeds.
Answer(s):
F1 = -0.311 lbf/ft and F2 = 0.414 lbf/ft
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Practice Problems on the Linear Momentum Equations
COLM_68
Consider a cart, with mass, M0, and an open top of area, A, rolling along a flat surface as shown in the figure below.
Rain, with a density, R, and falling vertically with speed, VR, with respect to the ground, is collected within the cart
through the cart’s open top. The water within the cart, with density, , discharges at an angle, , with respect to the
horizontal through a hole of area, Aout, located near the bottom of the cart. Note that the mass of water within the
cart does not remain constant.
rain with density, R, and vertical
speed¸VR, with respect to the ground
cross-section area, A
cart velocity, U, with
respect to the ground
g

discharge hole with
area, Aout
Neglecting drag forces and rolling friction, determine:
a. the rate at which water mass accumulates within the cart, i.e. dMH2O/dt, as a function of the current mass of
water within the cart, MH2O, and a subset of the following parameters: R, VR, U, A, , Aout, , M0, and g. You
do not need to solve any differential equations you derive. Hint: You may need to use Bernoulli’s equation in
this derivation.
b. the acceleration of the cart, dU/dt, in terms of a subset of the following parameters: R, VR, U, A, , Aout, , M0,
MH2O, and g. You do not need to solve any differential equations you derive.
Answer(s):
dM H 2O
2g 
 Aout
M H 2O   RVR A
dt
A
A
dU
M CV
   RUVR A  2 gM CV out cos 
dt
A
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Practice Problems on the Linear Momentum Equations
COLM_71
Water flows steadily through a reducing pipe bend as shown in the figure. The inlet and outlet conditions are
indicated on the diagram. Neglecting the weight of the bend and water, estimate the total force exerted by the flange
bolts.
1
D1 = 25 cm
p1 = 350 kPa (abs)
V1 = 2.2 m/s
patm = 100 kPa (abs)
D2 = 8 cm
p1 = 120 kPa (abs)
2
Answer(s):
 Fbolts  V12
 D12 
2
2
D   
 D    D12
1   1     p1,gage  p2,gage  2  
4   D2   
 D1   4

 
C. Wassgren, Purdue University
Page 46 of 49
Last Updated: 2010 Sep 16
Practice Problems on the Linear Momentum Equations
COLM_72
A cart, with an empty mass M0 and depth b into the page, travels up an incline (angle  with respect to the
horizontal). As the cart moves up the incline, it scoops a downward flow of water, with density , into the cart
interior as shown in the figure. The downward flow of water has a velocity profile approximated by:
Y
u U
h
where U is the velocity at the free surface (it’s constant) and h is the depth of the water layer (also constant).
Y
X
u U
Y
h
Vcart (rel. to ground)
g
scoop has depth b into the page
h
y
x

Determine the mass flux out of the cart in terms of (a subset of) , Vcart, U, h, b, and g.
Determine the x-momentum flux out of the control volume using the x-y frame of reference attached to the
control volume in terms of (a subset of) , Vcart, U, h, b, , and g.
Determine the net force acting in the x-direction on the control volume in terms of (a subset of) , Vcart, MCV, U,
h, b, , and g, where MCV is the instantaneous mass in the control volume.
Determine the acceleration of the cart in terms of (a subset of) , Vcart, MCV, U, h, b, , and g.
a.
b.
c.
d.
Answer(s):
 m out     12 U  Vcart  hb
2
  u x   u rel  dA    bh  13 U 2  VcartU  Vcart

CS
 Fnet, x   M CV g sin   12  g cos  h 2 b
2
2
2
1
1
dVcart  M CV g sin   2  g sin  h b   bh  3 U  VcartU  Vcart 


dt
M CV
C. Wassgren, Purdue University
Page 47 of 49
Last Updated: 2010 Sep 16
Practice Problems on the Linear Momentum Equations
COLM_73
A cart is propelled by water jet as shown in the figure below. The cart is filled with water to a height, h, has an
initial water mass of M0, a cart mass, Mcart, a cross-sectional area, A, and a (variable) velocity, V, with respect to the
ground. The water within the cart leaks through a small hole of area, Aexit, located in the bottom of the cart. The
hole area is much smaller than the cross-sectional area of the cart, i.e. Aexit << A. The propelling water jet has a nonuniform velocity profile given by:
  r 2 
u  U 1    
  R  
where U is the jet’s centerline velocity (with respect to the ground), R is the jet radius, and r is the radial distance
from the centerline of the jet. The wheels on the cart may be assumed frictionless and air drag may be neglected.
g
V
A
u(r) 2R
h
Aexit
a.
b.
c.
d.
Determine the speed of the water leaking through the bottom of the cart in terms of (a subset of) g, Aexit, and h.
Express the height of the water in the cart, h, in terms of the water mass within the cart, MH2O, the water’s
density, , and the cross-sectional area of the cart, A.
Determine the mass of the cart and water contained within, Mcart+MH2O, in terms of time, t, and (a subset of) M0,
Mcart, , g, A, Aexit, V, and U. Do not neglect the change in water mass within the cart.
Determine the acceleration of the cart in terms of (a subset of) your answer to part (c), , V, U, and R. You
need not solve any integrals that appear in your solution.
Answer(s):
V2  2 gh
h 
M H2 O
A
M cart+water  M cart  M H2 O  M cart
dV
2


dt M cart+water

g 
t
  M 0  Aexit
2 A 

2
2
   r 2 

r 0 U 1   R    V  rdr

 

r R
C. Wassgren, Purdue University
Page 48 of 49
Last Updated: 2010 Sep 16
Practice Problems on the Linear Momentum Equations
COLM_74
A pump in a tank of water (with a density of 1000 kg/m3 and dynamic viscosity 1.0*10-3 kg/(ms)) directs a jet at 10
m/s (relative to the cart) and with a volumetric flow rate of 1 m3/s against a vane as shown in the figure. At a given
instant in time, the cart speed is U = 10 m/s and the mass of the cart and the water contained within it is 200 kg.
Determine the acceleration of the cart at this instant in time if the jet follows:
a.
path A, or
b.
path B.
path B
g
60
path A
70
P
U
Answer(s):
dU

0
dt
dU
 QV cos 


dt
M CV
C. Wassgren, Purdue University
Page 49 of 49
Last Updated: 2010 Sep 16