Sheet No. (6)
Al-Azhar University.
Faculty of Engineering.
Rectilinear Motion
Engineering Mechanics.
Prep. Year Students.
(1) The motion of a particle is defined by the relation s = (t3 - 9t2 +24t – 2) where s is
expressed in meters and t in seconds. Determine
i)
the position, velocity, acceleration and the total distance traveled when t = 5
sec,
ii) when the velocity is zero,
iii) the position and the total distance traveled when the acceleration is zero.
(2) The acceleration of a particle is defined by the relation a = (64 – 12t2). The particle
starts at t = 0 with v = 0 and , s = 50 cm , Determine
i)
the time when the velocity is again zero,
ii) the position and velocity when t = 5 sec,
iii) the total distance traveled by the particle from t = 0 to t = 5 sec.
(3) A car is to be hoisted by elevator to the fourth floor of a parking garage, which is 12
m above the ground. If the elevator can accelerate at 0.18 m/s2 decelerate at 0.09
m/s2 and reach a maximum speed of 2.5 m/s, determine the shortest time to make
the lift, starting from rest and ending at rest.
(4) When a train is traveling along a straight track at 2 m/s, it begins to accelerate at a =
(60 v -4 ) m/s2. where v is in m/s .Determine its velocity v and the position 3 sec.
after the acceleration.
(5) A car starts from rest and moves along a straight line with an acceleration of a = (3
s -1/3 ) m/s2. where s is in meters. Determine the car’s acceleration when t = 4 sec.
(6) The acceleration of a rocket traveling upward is given by a = (6+ 0.02s) m/s2 where
s is in meters. Determine the time needed for the rocket to reach an altitude of s =
100 m. Initially, s = 0 and v = 0 when t = 0.
(7) A particle is moving along a straight line such that when it is at the origin it has a
velocity of v = 4 m/s. If it begins to decelerate at the rate of a = (-1.5 v 1/2 ) m/s2
where v is in m/s, determine the distance it travels before it stops and its position
and velocity when t = 2 sec.
(8) Ball A is released from rest at a height of 15
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m at the same time that a second ball B is
thrown upward 2 m from the ground. If the
balls pass one another at a height of 7.5 m,
determine the speed at which ball B was
thrown upward.
(9) When two cars A and B are next to one another, they are traveling in the same
direction with speeds vA and vB respectively. If B maintains its constant speed,
while A begins to decelerate at aA determine the distance d between the cars at the
instant A stops.
(10) If Both cars in problem 10 are traveling at 20 m/s when the driver of B suddenly
applies the brakes, causing his car to decelerate at 3.6 m/s2. It takes the driver of car
A 0.75 s to react (this is the normal reaction time for drivers). When he applies his
brakes, he decelerates at 4.8 m/s2. Determine the minimum distance d between the
cars so as to avoid a collision.
(11) The v–t graph for the motion of a
train as it moves from station A to
station B is shown. Draw the a–t
graph and determine the average
speed and the distance between the
stations.
(12) The s–t graph for a train has been
experimentally determined. From
the data, construct the v–t and a–t
graphs for the motion; 0 ≤ t ≤ 40 s.
For 0 ≤ t ≤30 s , the curve is a
parabola, and then it becomes
straight for t≥ 30 s.
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Sheet No. (7)
Al-Azhar University.
Faculty of Engineering.
Curvilinear Motion
Engineering Mechanics.
Prep. Year Students.
(1) The velocity of a particle is defined as v(t) = (0.8 t2 i + 12 t1/2 j +5k) m/s. determine
the magnitude and coordinate direction angles α, β, γ of the particle's acceleration
when t = 2 s.
(2) The position of particles A and B is described by rA = [2t i + ( t2 - l)j] m and rB = [(t
+ 2) i + (2t2 -5)j] m respectively, where t is in seconds. Determine the point where the
particles collide and their speeds just before the collision.
(3) The motion of a particle is defined by the equations x = t3 - 4t2 and y = t2 – 4t, where x
and y are expressed in meters and t in seconds. Determine
i)
the velocity and acceleration when t=1 sec, & t = 3 sec.
ii)
the time at which the value of the y coordinate is minimum, and the
corresponding velocity and acceleration of the particle.
(4) The motion of a vibrating particle is defined by the position vector r =[ (4 sin πt) i +
(cos 2πt) j ], where r is expressed in meters and t in seconds,
i)
Determine the velocity and acceleration when t = 1 sec.,
ii)
Show that the path of the particle is, parabolic.
(5) A particle moves in an elliptic path defined by the position vector r = (A cos pt)i + (B
sin pt)j. Show that the acceleration
(a) is directed toward the origin,
(b) is proportional to the distance; from the origin to the particle.
(6) The motion of a particle is defined by the equations (x =6t - 3 sin t) and (y= 6 -3 cos
t) , where x and t’ are expressed in meters and t is expressed in seconds. Sketch the
path of the particle for the time interval 0≤ t ≤ 2π, and determine (a) the magnitudes of
the smallest and largest velocities reached by the particle, (b) the corresponding times,
positions, and directions of the velocities.
(7) Water issues at A from a pressure tank with
a horizontal velocity V0. For what range of
values V0 will the water enter the opening
BC?.
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(8) A basketball player shoots when she is 5
m from the backboard. Knowing that the
ball has an initial velocity v at an angle of
30O with the horizontal, determine the
value of v when d is equal to (a) 228 mm,
(b) 430 mm.
(9) The girl always throws the toys at
an angle of 30° from point A as
shown. Determine the time
between throws so that both toys
strike the edges of the pool B and C
at the same instant. With what
speed must she throw each toy?
(10)The partial surface of the cam is that of a
logarithmic spiral r = (40 e0.05θ) mm, where
θ is in radians, If the cam is rotating at a

constant angular rate of  = 4 rad/s,
determine the magnitudes of the velocity and
acceleration of the follower rod at the instant
θ = 30°.
(11)Solve Prob. (10), if the cam has an angular

acceleration of  = 2 rad/s2 when its angular

velocity is  = 4 rad/s at = 30°.
(12)At the instant shown, the bicyclist at A is traveling
at 7 m/s around the curve on the race track while
increasing his speed at 0.5 m/s2. The bicyclist at B
is traveling at 8.5 m/s along the straight-a-way
and increasing his speed at 0.7 m/s2. Determine
the relative velocity and relative acceleration of A
with respect to B at this instant.
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Sheet No. (8)
Al-Azhar University.
Faculty of Engineering.
Kinetics of a Particle
Engineering Mechanics.
Prep. Year Students.
(1) A bar B of mass 300 kg, originally at rest, is
being towed over a series of small rollers.
Determine the force in the cable at time t= 5s if
the motor M is drawing in the cable for a short
time at a rate v = 0.4 t2. How far does the bar
move in time t? Neglect the mass of the cable,
pulley, and the rollers.
(2) The baggage truck A has mass 800 kg and is used to pull the two cars, each with mass
300 kg. The tractive force on the truck is F=480 N. Determine the initial acceleration of
the truck. What is the acceleration of the truck if the coupling at C suddenly fails? The
car wheels are free to roll. Neglect the mass of the wheels.
(3) A suitcase of weight 200 N slides from
rest a distance 5 m down the smooth ramp.
Determine the point where it strikes the
ground at C. How long does it take to go
from A to C?
(4) At the instant shown the block A of
weight 500 kg is moving down the plane at 2
m/s while being attached to the block B of
weight 250 kg. If the coefficient of kinetic
friction is μ k = 0.2, determine the
acceleration of A and the distance A slides
before it stops. Neglect the mass of the
pulleys and cables.
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(5) At the instant θ = 60 deg. the Girl’s center
of mass G has a downward speed 3m/s.
Determine the rate of increase in her speed and
the tension in each of the two supporting cords
of the swing at this instant. The boy has a
weight 300N. Neglect her size and the mass of
the seat and cords.
(6) If the girl’s center of mass G is momentarily at rest. Determine her speed and the
tension in each of the two supporting cords of the swing when θ = 90o. The boy has a
weight 300N. Neglect her size and the mass of the seat and cords.
(7) Determine the constant speed of the
passengers on the amusement-park ride if it
is observed that the supporting cables are
directed at angle θ =30o from the vertical.
Each chair including its passenger has a
mass 80 kg. Also, what are the components
of force in the n, t, and b directions which the
chair exerts on a passenger of mass 50 kg
during the motion?
(8) The snowmobile of mass 200 kg with
passenger is traveling down the hill at a
constant speed 6 m/s. Determine the
resultant normal force and the resultant
frictional force exerted on the tracks at the
instant it reaches point A. Neglect the size
of the snowmobile.
(9) The ball of mass 0.5 kg of negligible
size is fired up the vertical circular track
using the spring plunger. The plunger
keeps the spring compressed a distance δ
=0.08m when x = 0 . Determine how far x
it must be pulled back and released so that
the ball will begin to leave the track when
θ = 135 deg.
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(10) The block of weight 500 N slides down the
inclined plane for which the coefficient of kinetic
friction is μ k=0.25. If it is moving at speed 3m/s
when it reaches point A, determine the maximum
deformation of the spring needed to momentarily
arrest the motion. k = 3600 N/m
(11) The block of weight 25N is released
from rest at A and slides down the smooth
circular surface AB. It then continues to
slide along the horizontal rough surface
until it strikes the spring. Determine how
far it compresses the spring before
stopping.
(12) The block has a mass 0.8 kg and moves within the
smooth vertical slot. If it starts from rest when the
attached spring is in the unstretched position at A,
determine the constant vertical force F which must
be applied to the cord so that the block attains a
speed 2.5 m/s when it reaches 0.15 m. Neglect the
size and mass of the pulley.
Hint: The work of F can be determined by finding the
difference Δl in cord lengths AC and BC and using
UF = F Δl
(13) The cyclist travels to point A, pedaling
until he reaches speed 8m/s. He then coasts
freely up the curved surface. Determine the
normal force he exerts on the surface when
he reaches point B. The total mass of the
bike and man is 75 kg. Determine also how
high he reaches up the surface before he
comes to a stop. And, what are the resultant
normal force on the surface at this point and
his acceleration? Neglect friction, the mass
of the wheels, and the size of the bicycle.
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