1. Fundamentals of Physics School of Physical Science and Technology
Mechanics
(Bilingual Teaching)
张昆实
School of Physical Science and Technology
Yangtze University

2. Chapter 10 Collisions
10-1 What is a Collision?
10-2 Impulse and Linear Momentum
10-3 Momentum and Kinetic Energy in
Collisions
10-4 Inelastic Collisions in One
Dimension
10-5 Elastic Collisions In One
10-6 Collisions in Two Dimensions

3. 10-1 What is a Collision? In everyday language, a collision occurs
when an objects crash into each other.
Exp: (a) Meteor Crater (流星坑)
(b) particles scattering (粒子散射)
(c) tennis ball contacts with racket
A collision is an isolated event in which two or more bodies (the colliding bodies) exert relatively strong forces on each other for a relatively short time definition
Note : a collision does not require contact,
a collision force does not have to be a
force involving contact.

4. 10-2 Impulse and Linear Momentum
Single collision
During a head-on collision:
A third law force pair ,
The change of the linear momentum depends on:
The forces and the action time
Apply Newton’s second law to ball R
(10-1)
Integrating Eq over the interval :
(10-2)

5. 10-2 Impulse and Linear Momentum
(10-2)
(10-3)
is the impulse, which is a measure of both the strength and the duration of the collision force. J
rectangle
(10-8)

6. 10-2 Impulse and Linear Momentum
(10-2)
(10-3)
From Eq and 10-3
(10-4)
( Impulse-linear momentum theorem )
Component form
(10-5)
(10-6)
(10-7)

7. 10-2 Impulse and Linear Momentum
Series of collisions
Projectiles
Target
A steady stream of projectile
bodies ( ), moves along
axis and collides with a
fixed target. Find the average
force on the target during the bombardment.
The total change in momentum for
projectiles in is :
The impulse on the target in :
(10-9)
Combining Eq and 10-9:
(10-10)

8. 10-2 Impulse and Linear Momentum
Series of collisions
Projectiles
Target
The rate the projectiles
collide with the target
(10-10)
In , an amount of mass collide with the target, so
(10-13)
The rate the mass
collides with the target If If

9. 10-3 Momentum and Kinetic Energy in Collisions
discussion collisions in closed (no mass enters
or leaves them) and isolated (no net external
forces act on the bodies within them ) systems.
Kinatic Energy（in collisions）
1. Elastic collision: is a special type of collision in which the kinetic energe of the system of colliding bodies is conserved.
2. Inelastic collision: after the collision the kinetic energe of the system is not conserved.
3. Completely inelastic collision: If the bodies stick together and have the same final velocity after the collision.

10. 10-3 Momentum and Kinetic Energy in Collisions
Linear Momentum（in collisions）
Regardless of the details of the impulses in a collision; Regardless of what happens to the kinetic energy of the system, The total linear momentum of a closed, isolated systems cannot change. (no external force !)
The law of conservation of linear momentum
In a closed , isolated system containing a collision, the linear momentum of each colliding body may change but the total linear momentum of the system cannot change, whether the collision is elastic or inelastic.

11. 10-4 Inelastic Collisions in One Dimension
befor
after
One dimensional collision
Two colliding bodies form a closed , isolated system
along the x axis.
The law of conservation of linear momentum
(10-15)
before、 after collision
(10-16)
The motion is one-dimensional, component form:

12. 10-4 Inelastic Collisions in One Dimension
Completely Inelastic collision
after collision
befor
Projectile
Target
Before the collision body 2
is at rest, body 1 moves directly toward it. After the collision, the stuck-together bodies move with the same velocity .
The law of conservation of linear momentum
(10-17)
before、 after collision
(10-18)

13. 10-4 Inelastic Collisions in One Dimension
Velocity of Center of Mass
Collision !
In a closed, isolated system, the velosity of the center of mass of the system cannot be changed by a collision for there is no net external force to change it.
Find out
(10-19)
(10-20)
Example
(10-21)

14. 10-5 Elastic Collisions In One Dimension
after
befor
Projectile
Target
Stationary Target
An elastic collision is a special type of collision in which the kinetic energy of the system of cilliding bodies is conserved.
Total kinetic energy
before the collision
after the collision = ☞
In an elastic collision, the kinetic energy of each colliding body may change, but the total kinetic energy of the system does not change.

15. 10-5 Elastic Collisions In One Dimension
after
befor
Projectile
Target
Stationary Target
Before the collision body 2
is at rest, body 1 moves toward it (head on collision).
Assume this two body system
is closed and Isolated. Then the
net linear momentum of
the system is conserved.
( linear momentum )
(10-26) ☞
If the collision is also elastic, the total kinetic energy of the system is also conserved.
(10-27)
( kinetic energy )

16. 10-5 Elastic Collisions In One Dimension
Stationary Target
( linear momentum )
(10-26)
(10-27)
( kinetic energy )
If know , Then can find out
(10-28)
Rewrite Eq.10-26
(10-29)
Rewrite Eq.10-27
(10-30)
(10-31)

17. 10-5 Elastic Collisions In One Dimension
after
befor
Projectile
Target
Stationary Target
(10-30)
(10-31)
discussion
1. From (10-30) , ball 2 always moves forward
2. From (10-31), if
ball 1 moves forward
ball 1 bounds back If

18. 10-5 Elastic Collisions In One Dimension
Stationary Target
(10-30)
(10-31)
discussion
A few special situations
1. Equal masses: If Eqs. (10-30) and (10-31)
reduce to
and
Changing vilocities !
2. A massive target: If Eqs. (10-30) and (10-31)
reduce to
and
(10-32)
bounds back
3. A massive projectile: If Eqs. (10-30) and (10-31)
reduce to
and
(10-33)

19. 10-5 Elastic Collisions In One Dimension
1.Conservation of the linear momentum
Moving Target
2.Conservation of the kinetic energy
head on elastic collision
(10-36)
To work out and rewrite these conservation equations:
(10-37)
(10-38)
(10-39)

20. 10-6 Collisions in Two Dimensions
glancing collision
Two dimension (not head-on)
elastic collision
System: closed + isolated
Conservation equations:
(10-40)
(10-41)
Rewrite Eq for components along the x、y axis
(10-42)
(10-43)
(10-44)