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Collisions and Momentum
• Conservation of Momentum
• Elastic and inelastic collisions
Serway & Jewett 9.3 - 9.5
Physics 1D03 - Lecture 26
Collisions
A collision is a brief interaction between two
(or more) objects. We use the word “collision”
when the interaction time Δt is short relative to
the rest of the motion.
During a collision, the objects exert equal and
opposite forces on each other. We assume
these “internal” forces are much larger than
any external forces on the system.
We can ignore external forces if we compare
velocities just before and just after the
collision, and if the interaction force is much
larger than any external force.
v1,i
m1
v2,i
m2
F1
F2 = -F1
v1,f
v2,f
Physics 1D03 - Lecture 26
Elastic and Inelastic Collisions
Momentum is conserved in collisions. Kinetic energy is sometimes
conserved; it depends on the nature of the interaction force.
A collision is called elastic if the total kinetic energy is the same
before and after the collision. If the interaction force is
conservative, a collision between particles will be elastic (eg:
billiard balls).
If kinetic energy is lost (converted to other forms of energy), the
collision is called inelastic (eg: tennis ball and a wall).
A completely inelastic collision is one in which the two colliding
objects stick together after the collision (eg: alien slime and a
spaceship). Kinetic energy is lost in this collision.
Physics 1D03 - Lecture 26
Quiz
In dense regions of galaxies, or when galaxies
collide, the stars are said to collide. Even though they
actually do not hit each other, what type of collision
do they undergo:
a) elastic
b) inelastic
c) depends on whether they form a binary star or not
Physics 1D03 - Lecture 26
If there are no external forces, then the total momentum is
conserved:
v1,i
m1
v2,i
m2
p1,i + p2,i = p1,f + p2,f
v1,f
v2,f
This is a vector equation. It applies to each component of p
separately.
Physics 1D03 - Lecture 26
Elastic Collisions
In one dimension (all motion along the x-axis):
1) Momentum is conserved:
m1v1i  m2 v2i  m1 v1f  m2v2f
In one dimension, the velocities are represented by positive
or negative numbers to indicate direction.
2) Kinetic Energy is conserved:
1
2
m1v12i  12 m2v22i  12 m1v12f  12 m2v22f
We can solve for two variables if the other four are known.
Physics 1D03 - Lecture 26
One useful result: for elastic collisions, the magnitude of the relative
velocity is the same before and after the collision:
|v1,i – v2,i | = |v1,f – v2,f |
(This is true for elastic collisions in 2 and 3 dimensions as well).
An important case is a particle directed at a stationary target (v2,i = 0):
• Equal masses: If m1 = m2, then v1,f will be zero (1-D).
• If m1 < m2, then the incident particle recoils in the opposite direction.
• If m1 > m2, then both particles will move “forward” after the collision.
before
after
Physics 1D03 - Lecture 26
Elastic collisions, stationary target (v2,i = 0):
Two limiting cases:
1) If m1 << m2 , the incident particle
rebounds with nearly its original
speed.
v1
-v1
2) If m1 >> m2 , the target particle moves
away with (nearly) twice the original
speed of the incident particle.
v1
v1
2v1
Physics 1D03 - Lecture 26
Quiz
A tennis ball is placed on top of a basketball and both are dropped.
The basketball hits the ground at speed v0. What is the maximum
speed at which the tennis ball can bounce upward from the
basketball? (For “maximum” speed, assume the basketball is
much more massive than the tennis ball, and both are elastic).
a) v0
b) 2v0
c) 3v0
?
v0
v0
Physics 1D03 - Lecture 26
Example – inelastic collision:
A neutron, with mass m = 1 amu (atomic mass unit),
travelling at speed v0, strikes a stationary deuterium
nucleus (mass 2 amu), and sticks to it, forming a
nucleus of tritium. What is the final speed of the
tritium nucleus?
Physics 1D03 - Lecture 26
An elastic collision:
Two carts moving in opposite direction collide and bounce back.
If cart 1 bounces back with v=2m/s, what is the speed of cart 2 ?
6 m/s
2 kg
5 m/s
4 kg
Physics 1D03 - Lecture 26
Example
A 1500kg car traveling east with a speed of 25m/s
collides with a 2500kg car traveling north at a speed
of 20m/s. Find the direction and velocity of the cars
after the collision, assuming that they stick together.
Physics 1D03 - Lecture 26
Summary
• Momentum is conserved in collisions.
• In elastic collisions, kinetic energy is also conserved.
Physics 1D03 - Lecture 26