Download Lecture 26 - McMaster Physics and Astronomy

Collisions
• Conservation of Momentum
• Elastic and inelastic collisions
Serway 9.3 - 9.4
For practice: Chapter 9, problems 10, 11, 23, 70, 75
Physics 1D03
Momentum: p = mv
Impulse (a vector) is defined as F ∆t (for a constant force),
∫ F dt
or
in general.
Newton’s Second Law again:
F =
dp
dt
or, alternatively,
(change in p) = (total impulse from external forces)
Physics 1D03
Newton #3 and Momentum Conservation
Two particles interact:
F1
∆p1 = F1 ∆t
∆p2= F2 ∆t = -F1 ∆t = -∆p1
m1
F2 = -F1
m2
The momentum changes are equal and opposite.
The total momentum of the system (two particles) doesn’t change:
p = p1 + p2 ; ∆p = ∆p1 + ∆p2 = 0
Total momentum p is constant
The fine print: only internal forces act. External forces would
transfer momentum into or out of the system.
Physics 1D03
Momentum Conservation
Newton’s 3rd Law: F2 = -F1
leads to the
Law of Conservation of Momentum:
F1
m1
F2 = -F1
m2
The total momentum (vector sum) of all the
particles in a system remains constant if there are
no external forces acting on the system.
Physics 1D03
Example: roller basketball
A (stationary) 100-kg basketball player on roller skates shoots a
1-kg basketball horizontally at a speed of 10 m/s. What happens
(if there is no friction)?
Is momentum still conserved after an opponent
catches the ball and holds it?
What happens if the ball is thrown upward at an angle
of 37 degrees to the floor?
Physics 1D03
Quiz
A kaon ( a strange type of subatomic
particle) at rest spontaneously splits into
three other particles, called π+, π−, and π0,
which fly off in different directions. The π+
and π− have equal masses, but the π0
mass is different.
The velocities of two particles are shown;
in which direction is the third one moving?
2v
π−
v
π+
A)
B)
C)
D)
E) We can’t tell exactly without knowing its mass
Physics 1D03
Collisions
A collision is a brief interaction between two
(or more) objects.
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.
v1,i
m1
v2,i
m2
F1
F2 = -F1
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,f
v2,f
Physics 1D03
If there are no external forces, then the total momentum is
conserved:
p1,i + p2,i = p1,f + p2,f
This will be a good approximation if the impulse from any
external forces is small.
This is a vector equation. It applies to each component of p
separately.
Physics 1D03
Example:
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?
What does the energy balance look like?
Physics 1D03
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
If kinetic energy is lost (converted to other forms of energy), the
collision is called inelastic.
A completely inelastic collision is one in which the two colliding
particles stick together after the collision.
Question: Why not define “completely inelastic” to mean the final
kinetic energy is zero?
Physics 1D03
Quiz
When two stars moving through the Galaxy pass close to each
other, their mutual gravitational pull causes them to change
speed and direction. We can call this a “collision”, even though
there was no contact between the stars. Will this be an elastic
or inelastic collision?
a) elastic or nearly elastic
b) Inelastic
Physics 1D03
An elastic collision in slow motion:
If k = 1200 N/m, what is the maximum compression of the spring
during the collision?
6 m/s
2 kg
3 m/s
4 kg
Is the momentum constant at all times?
Is the kinetic energy constant at all times?
Is the mechanical energy constant at all times?
Physics 1D03
4 kg
4 kg
QUIZ
(no friction, ideal spring,…)
Which quantities do not
remain constant at all
times during the collision?
4 kg
A) Momentum only
B) Kinetic energy only
C) Mechanical energy only
D) All are constant
E) Two are not constant
4 kg
4 kg
Physics 1D03
An elastic collision in slow motion:
If k = 1200 N/m, what is the maximum compression of the spring
during the collision?
6 m/s
2 kg
3 m/s
4 kg
Physics 1D03