1.3a Elastic and inelastic collisions
Momentum
The phrase ‘gathering momentum’ is used in everyday life to indicate that
something or someone has got going and is likely to prove difficult to stop.
There are echoes of this in sport, where the phrase ‘the momentum has turned in
the other team’s favour’ can often be heard. In Physics momentum is an
indication of how difficult it would be to stop something. The faster or more
massive an object is the more momentum it will have. The precise definition of
momentum is:
The momentum (p) of any object is the product of its mass ( m) and its velocity
(v):
p  mv
Since mass is measured in kilograms and velocity in metres per second, the units
of momentum are kg m s –1 .
Momentum is a vector quantity, and the direction of the momentum is the same
as the direction of the velocity. It is useful because it is a conserved quantity, ie
the total momentum is the same before and after a collision, in the absence of
external forces. This is called the principle of conservation of momentum.
Momentum in collisions
Notes
1.
You should learn the statement of the principle of conservation of
momentum: the total quantity of momentum before a collision is the same
as the total quantity of momentum after the collision in the absence of an
external force.
2.
This is a fundamental law of physics and applies to all collisions: road
accidents, collisions between meteors and planets, collisions between
atoms.
3.
The law applies to total momentum, not individual momentum.
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4.
Since momentum is a vector quantity we cannot add momenta (plural of
momentum) like ordinary numbers; we must take account of direction. For
the problems that we will consider this means that some momenta (usually
in the original direction) may be positive (+) while other momenta (the
opposite direction) are negative (–).
5.
In problems it is essential to demonstrate that you know the conservation
law. This should be stated as part of your working as:
Total momentum before collision = total momentum after collis ion
Worked examples
1.
Two cars are travelling towards each other as shown below. They collide,
lock together and move forwards (ie to the right) after the collision.
Find the speed of the cars immediately after the collision.
Before
After
10 m s–1
8 m s–1
1200 kg
A
1000 kg
B
?
1200 kg
1000 kg
Take motion  as +
m 1 u 1 = 1200 × 10 = 12,000
(m 1 + m 2 )v = (1200 + 1000)v
m 2 u 2 = 1000 × –8 = - 8000
= 2200v
total momentum before = total momentum after
12,000 – 8000 = 2200v
v = 4000 = 1.8
2200
ie = 1.8 m s –1 to the right
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2.
One vehicle (vehicle A) approaches another (vehicle B) from behind as
shown below. The vehicles are moving with the speeds shown. After the
collision the front vehicle is travelling at 11 m s –1 . Calculate the speed of
vehicle B after the collision.
Before A
B
12 m s–1
A
9 m s–1
1200 kg
After
B
11 m s–1
800 kg
1200 kg
?
800 kg
Take motion  as +
m 1 u 1 = 1200 × 12 = 14,400
m 1 v 1 = 1200 × v 1 11
m 2 u 2 = 800 × 9 = 7200
m 2 v 2 = 800 × v 2
total momentum before = total momentum after
14,400 + 7200 = 1320 + 800v 2
21,600 = 1320 + 800v 2
v2 
20,280
 25.35
800
ie v 2 = 25.4 ms –1 to the right
Note: In momentum problems it can help to lay out your working under the
headings ‘before’ and ‘after’. Always draw a diagram of the situation before and
after, including all relevant details such as masses, velocities and directions of
motion. Include the statement of conservation of momentum.
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Kinetic energy in collisions
Momentum is always conserved in collisions and explosions. By the law of
conservation of energy, the total energy is also conserved in collisions and
explosions, but kinetic energy is not necessarily conserved.
There are two kinds of collision:
(a)
those in which kinetic energy (KE) is conserved
ie total KE before = total KE after
This is called an elastic collision.
(b)
those in which kinetic energy is not conserved
ie KE is lost during the collision to other forms of energy, such as heat
energy
This is called an inelastic collision.
If after a collision the objects stick together, this is always an inelastic collision.
If the objects bounce apart the collision may be elastic; the only sure way of
finding out is to calculate the total KE before and after the collision. Usually this
will involve using conservation of momentum first to calculate all the relevant
velocities. Remember, momentum is always conserved in the absence of external
forces.
Reminder: E K  1 mv 2
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A special case
If two objects of the same mass collide elastically, they exchange velocities after
the collision. For example, if a cue ball travelling at 2 m s –1 collides with a
second snooker ball of the same mass (a head-on collision, with no spin
involved), the second ball will move off at 2 m s –1 and the cue ball will stop.
This effect can be seen in ‘Newton’s cradle’.
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