Examples of momentum
Look at this example.
A red and a blue snooker ball both have a mass of 160 g. They collide with speeds and directions as shown in this diagram.
The following steps show you how to calculate the velocity of the red ball after the collision, and use kinetic energy to determine whether the collision was elastic or not.
\(\text{total momentum before} = {\text{total momentum after}}\)
\({\text{m}}_{\text{red}}{\text{u}_{\text{red}}} + {\text{m}}_{\text{blue}}{\text{u}_{\text{blue}}} = {\text{m}}_{\text{red}}{\text{v}_{\text{red}}} + {\text{m}}_{\text{blue}}{\text{v}_{\text{blue}}}\)
\(0.16 \times 0.28 + 0.16 \times ( - 0.12) = 0.16 \times {\text{v}}_{\text{red}} + 0.16 \times 0.18\)
Rearranging, ie right to left.
\(0.16 \times{\text{v}}_{\text{red}} = (0.0448 - 0.0192) - 0.0288\)
\(= 0.0256 - 0.0288\)
\(= \frac{{ - 0.0032}}{{0.16}}\)
\({\text{v}}_{\text{red}}~=~- 0.02{\text{ ms}}^{-1}\)
To determine if the collision is elastic or not, you must work out the kinetic energy before and after the collision.
Remember the equation for calculating kinetic energy in general.
\(\text{E}_{\text{k}}=\frac{1}{2}{\text{mv}^{2}}\)
This example deals with two objects. The kinetic energy before the collision involves their initial speed, so the equation becomes
\(\text{E}_{\text{k}}=\frac{1}{2}{\text{m}}_{\text{red}}{{\text{(u}}_{\text{red}}{\text{)}}}^{2}+\frac{1}{2}{\text{m}}_{\text{blue}}{{\text{(u}}_{\text{blue}}{\text{)}}}^{2}\)
\(= 0.5 \times 0.16 \times {(0.28)^2} + 0.5 \times 0.16 \times {(0.12)^2}\)
\(= 0.00627 + 0.00115\)
\(= 0.00742{\text{ J}}\)
Total kinetic energy after collision
\(\frac{1}{2}{\text{m}}_{\text{red}}{{\text{(v}}_{\text{red}}{\text{)}}}^{2}+\frac{1}{2}{\text{m}}_{\text{blue}}{{\text{(v}}_{\text{blue}}{\text{)}}}^{2}\)
\(=0.5\times0.16\times(0.02)^2+0.5\times0.16\times(0.18)^2\)
\(=0.000032+0.00259\)
\(=0.00262{\text{ J}}\)
Total kinetic energy lost
\(=0.00742-0.00262\)
\(=0.0048{\text{ J}}\)
48 mJ of energy is lost so the collision is inelastic.
Question
If two bumper cars collide head-on in a fairground and both cars come to a stop due to the collision, kinetic energy is obviously not conserved. Is momentum conserved even though both cars stop?
Yes, although there is no momentum after the collision, there was no total momentum before the collision. The positive momentum of one car must have been balanced out by the negative momentum of the other car.
The total momentum of two objects before a collision is the same as their total momentum after the collision - provided there are no external forces.