Chapter 11
Kinematics of Particles
11.1 Introduction to Dynamics
Galileo and Newton (Galileo’s experiments led to Newton’s laws)
Kinematics and dynamics (kinetics)
(Dynamics is composed of kinematics and kinetics.)
Rectilinear Motion of Particles
11.2 Position, Velocity, and Acceleration
v
x
t
(units of m/s, ft/s, etc.)
x dx

t 0 t
dt
v  lim
a
v
t
(units of m/s2, ft/s2, etc.)
v dv d 2 x


t 0 t
dt dt 2
a  lim
Also
a
dv dv dx

or
dt dx dt
av
dv
dx
Jerk 
da
dt
Figure 11.6
11.3 Determination of the Motion of a Particle
Three common classes of motion
1. a  f (t )
dv  adt  f (t )dt
t
v  v0   f ( t )dt 
0
dx
 v0
dt
t t


x  x0  v0 t     f ( t )dt dt
0 0

2. a  f ( x)  v
dv
dx
vdv  adx  f ( x)dx
x
1
2
 f ( x)dx
(v 2  v02 ) 
xo
and v 
dx
dt
then get x  x(t )
3. a  f (v) 
v

v0
dv
dt
t
dv
  dt  t
f (v ) 0
for a  v
x
v
x0
v0
dv
 f (v)
dx
vdv
 dx   f (v)
Both can lead to x  x(t )
11.4 Uniform Rectilinear Motion
v  constant
a 0
x  x0   vdt  vt
x  x0  vt
11.5 Uniform Accelerated Rectilinear Motion
a  constant
v  v0  at
x  xo  v0 t  12 at 2
Also v
dv
a
dx
v 2  v02  2a ( x  x0 )
11.6 Motion of Several Particles
When independent particles move along the same line, independent equations exist
for each.
Use the same origin and time.
Relative motion of two particles
xB A  xB  x A the relative position of B with respect to A
vB A  vB  v A the relative velocity of B with respect to A
aB A  aB  a A the relative acceleration of B with respect to A
Dependent motions
See Figure 11.8
System has one degree of freedom since only one coordinate can be chosen
independently.
Figure 11.9 has 2 degrees of freedom
11.7 Graphical Solution of Rectilinear – Motion Problems
v
dx
 slope of x  t curve
dt
a
dv
 slope of v  t curve
dt
t2
x2  x1   vdt is the change in x equal to the area under the v-t curve.
t1
t2
v2  v1   adt is the change in v equal to the area under the a-t curve.
t1
If a  constant then v is linear in t and x is quadratic in t.
11.8 Other Graphical Methods
See Figure 11.12
v1
x1  x0  v0 t 1  (t1  t )dv
dv  adt
v0
t1
x1  x0  v0t 1  (t1  t )adt
0
Note
t1
t1
0
0
 tadt  t  adt  t (area under a  t curve)
Moment-area method
x1  x0  v0t 1(area under a  t curve)(t1  t )
Consider v-x curve. See Figure 11.13
BC  AB tan  v
dv
a
dx
Curvilinear Motion of Particles
11.9 Position Vector, Velocity, and Acceleration
See Figures 11.??? And 11.???



r d r
v  lim

t 0 t
dt
v
ds
dt

 dv
a
dt
11.10 Derivatives of Vector Functions


dP
P
 P( u  u )  P( u ) 
 lim
 lim 

du u0 u u0 
u

 

d ( P  Q ) dP dQ


du
du du


d ( fP ) df 
dP

P f
du
du
du

 

d ( P  Q ) dP   dQ

Q  P
du
du
du

 

d ( P  Q ) dP   dQ

Q  P
du
du
du

dPy
dP dPx
ˆj  dPz k̂

î 
du du
du
du
Rate of Change of a Vector

P  Px î  Py ˆj  Pz k̂
See italics at bottom of section
11.11 Rectangular Components of Velocity and Acceleration

r  xî  yˆj  zk̂

v  xî  yˆj  zk̂

a  xî  yˆj  zk̂
See projectile example to show how equation can be separated into parts in some
cases.
11.12 Motion Relative to a Frame in Translation
  
rB  rA  rB A
  
rB  rA  rB A

 
vB  v A  vB A



aB  a A  aB A
11.13 Tangential and Normal Components
Of acceleration with respect to the path of motion
Plane Motion of a Particle
Note what happens if we do
êt
êt
 sin(  2 ) 
 2 sin(  2 ) 
 ên lim
 ên lim 

ê
lim
n
  ên
 0 
 0 
 0 
 0 
 
  2 
lim

v  vêt

dê
 dv dv
a
 êt  v t
dt dt
dt
Remember that
s  r
or
s
 0 
r  lim
dêt dêt d ds v

 ên
dt d ds dt 
 dv
v2
a  êt  ên
dt

Discuss changing radius of curvature for highway curves
Motion of a Particle in Space
11.14 Radial and Transverse Components
Plane motion
dêr
 ê
d
dê
 êr
d
Note êr  î cos  ˆj sin 
dêr
 î sin   ˆj sin   ê
d
dêr dêr d 

  ê
dt
d dt

 dr
v
 rêr  rêr  rêr  r ê  vr êr  v ê
dt
vr  r
v  r

a  ( r  r 2 )êr  ( r  2r )ê
ar  r  r 2
Note a r 
a  r  2 r
dv
dvr
and a  
dt
dt
Extension to the Motion of a Particle in Space: Cylindrical Coordinates

r  Rêr  zk̂

v  R êR  Re  zk̂

  R 2 )ê  ( R  2 R  )ê  zk̂
a (R
R
