Explosions

The conservation of energy and momentum is very useful in the study of the motion of objects after an explosion. The same ideas can be applied to firing a rifle, a radium nucleus emitting an alpha particle and a bow firing an arrow etc.

Kinetic energy in explosions

Consider an explosion, such as a cannon firing a shell, that produces two fragments of masses m (shell) and M (cannon) with velocities u and v respectively. (See Figure 1).



By the law of conservation of momentum the two objects must move off in opposite directions.

Let the kinetic energy of the shell, mass m, be E and that of the cannon, mass M, be E'.

Now:  E = ½ mu² and E' = ½ Mv²

By the conservation of momentum mu = -Mv and therefore:

E= ½ mu² and E' = ½ [m²u²]/M

This gives:



and so the fragment with the smaller mass has the larger kinetic energy.

A bullet with a mass one hundredth that of the gun that fires it will have one hundred times the kinetic energy of the gun.



This principle can be studied with two spring loaded trolleys (Figure 2). If they are put back to back and the spring in one is released they will explode. Both trolleys move apart with the same speed because they are of equal mass. In the second diagram trolley 2 moves off with half the speed of trolley 1 as it has twice the mass.



An interesting case is where no particle is emitted – e.g. the bow is bulled back under tension and then released with no arrow being fired. There is a danger of breaking the bow since all the energy stored in the stretched string goes into the motion of the bow. You can test this by throwing a ball and then doing the throwing action again but this time without a ball in your hand – it hurts your arm as all the stored energy must be absorbed by your arm again in bringing it to rest.

Relative velocity in collisions

It is often very useful to use the idea of relative velocity in a collision problem. One of the objects involved is considered to be at rest, the relative velocity of the other is found and then the final velocities of the objects relative to the frame of reference can be calculated.

The following example demonstrates the use of relative velocity in a collision problem.