HIGHER TECHNOLOGICAL INSTITUTE (HTI)
Mechatronics Department
Mechanics 2
Dr. Tarek Abbas
Lec #3
Acceleration as a function of time,
position, or velocity
If….
a  a t 
a  a  x
Kinematic relationship
dt 
dx
dv
and a 
v
dt
dv
 a (v )
dt
dv
v  a v
dx
v
t
v0
0
 dv   a  t  dt
dv
 a(t )
dt
v dv  a  x  dx
a  a v
Integrate
v
x
v0
x0
 v dv   a  x  dx
v
t
dv
v a  v   0 dt
0
x
v
v dv
x dx  v a  v 
0
0
Uniformly Accelerated Rectilinear
Motion For a particle in uniformly accelerated rectilinear
motion, the
acceleration of the particle is constant. You may recognize these
constant acceleration equations from your physics courses.
dv
 a  constant
dt
dx
 v0  at
dt
x
v
t
 dv  a  dt
v0
0
t
 dx    v0  at  dt
x0
dv
v  a  constant
dx
v  v0  at
x  x0  v0t  12 at 2
0
v
x
v0
x0
 v dv  a  dx
v 2  v02  2a  x  x0 
Careful – these only apply to uniformly
accelerated rectilinear motion!
11 - 3
Motion of Several Particles: Relative
• For particles moving along the same line, time
Motion
should be recorded from the same starting
instant and displacements should be measured
from the same origin in the same direction.
 x B  x A  relative position of B
with respect to A
xB  x A  xB A
xB
A
 v B  v A  relative velocity of B
with respect to A
vB  v A  vB A
vB
A
 a B  a A  relative acceleration of B
with respect to A
aB  a A  aB A
aB
A
11 - 4
Motion of Several Particles: Dependent
Motion
• Position of a particle may depend on position of one
or more other particles.
• Position of block B depends on position of block A.
Since rope is of constant length, it follows that sum of
lengths of segments must be constant.
x A  2 x B  constant (one degree of freedom)
• Positions of three blocks are dependent.
2 x A  2 x B  xC  constant (two degrees of freedom)
• For linearly related positions, similar relations hold
between velocities and accelerations.
dx
dx A
dx
 2 B  C  0 or 2v A  2v B  vC  0
dt
dt
dt
dv
dv
dv
2 A  2 B  C  0 or 2a A  2a B  aC  0
dt
dt
dt
2
11 - 5