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2 Particle Dynamics
Elastic Collisions (One-Dimensional Head-On)
By definition total kinetic energy is conserved. If u 1 and u 2 be the respective initial
velocities of m 1 and m 2 , v 1 and v 2 being the corresponding final velocities, then
u 1 − u 2 = v 2 − v 1
(2.12)
Thus, the relative velocity of approach before the collision is equal to the relative
velocity of separation after the collision:
v 1 =
m 1 − m 2
m 1 + m 2
u 1 +
2m 2 u 2
m 1 + m 2
(2.13)
v 2 =
2m 1 u 1
m 1 + m 2
+
m 2 − m 1
m 1 + m 2
u 2
(2.14)
Inelastic Collisions, Direct Impact
The bodies stick together in the course of collision and are unable to separate
out. After the collision they travel as one body with common velocity v given by
v =
m 1 u 1 + m 2 u 2
m 1 + m 2
(2.15)
Energy wasted =
1
2
μ(u 1 − u 2 )
2
(2.16)
where μ is the reduced mass.
Ballistic pendulum is a device for measuring the velocity of a bullet. The pendulum consists of a large wooden block of mass M which is supported vertically
by two cords. A bullet of mass m hits the block horizontally with velocity v and
is lodged within it. As a result of collision the block is raised through maximum
height h (see prob. 2.44). Applying momentum conservation for the initial collision
process, and energy conservation for the subsequent motion, it can be shown that
v =
1 +
M
m
2gh
(2.17)
Partially elastic collisions are collisions which fall in between perfectly elastic collisions and totally inelastic collisions. The coefficient of restitution e which
defines the degree of inelasticity is given by
e = −
relative velocity of separation
relative velocity of approach
=
v 1 − v 2
u 2 − u 1
(2.18)
For perfectly elastic collisions e = 1, for totally inelastic collisions e = 0 and for
partially elastic collisions 0 < e < 1.
2 Particle Dynamics
Elastic Collisions (One-Dimensional Head-On)
By definition total kinetic energy is conserved. If u 1 and u 2 be the respective initial
velocities of m 1 and m 2 , v 1 and v 2 being the corresponding final velocities, then
u 1 − u 2 = v 2 − v 1
(2.12)
Thus, the relative velocity of approach before the collision is equal to the relative
velocity of separation after the collision:
v 1 =
m 1 − m 2
m 1 + m 2
u 1 +
2m 2 u 2
m 1 + m 2
(2.13)
v 2 =
2m 1 u 1
m 1 + m 2
+
m 2 − m 1
m 1 + m 2
u 2
(2.14)
Inelastic Collisions, Direct Impact
The bodies stick together in the course of collision and are unable to separate
out. After the collision they travel as one body with common velocity v given by
v =
m 1 u 1 + m 2 u 2
m 1 + m 2
(2.15)
Energy wasted =
1
2
μ(u 1 − u 2 )
2
(2.16)
where μ is the reduced mass.
Ballistic pendulum is a device for measuring the velocity of a bullet. The pendulum consists of a large wooden block of mass M which is supported vertically
by two cords. A bullet of mass m hits the block horizontally with velocity v and
is lodged within it. As a result of collision the block is raised through maximum
height h (see prob. 2.44). Applying momentum conservation for the initial collision
process, and energy conservation for the subsequent motion, it can be shown that
v =
1 +
M
m
2gh
(2.17)
Partially elastic collisions are collisions which fall in between perfectly elastic collisions and totally inelastic collisions. The coefficient of restitution e which
defines the degree of inelasticity is given by
e = −
relative velocity of separation
relative velocity of approach
=
v 1 − v 2
u 2 − u 1
(2.18)
For perfectly elastic collisions e = 1, for totally inelastic collisions e = 0 and for
partially elastic collisions 0 < e < 1.