User:Inconspicuum/Physics (A Level)/Conservation of Momentum

Momentum is the product of the mass of an object and its velocity. It is usually denoted p:

$$p = mv$$,

where m is mass, and v is velocity. The total momentum in a closed system is always conserved. This fact is useful, since it allows us to calculate velocities and masses in collisions.

Collisions
Let us consider a basic example: a ball of mass M collides with velocity u with a stationary ball of mass m. The stationary ball has no momentum before the collision, and the moving ball has momentum Mu. This must equal the momentum of both balls after the collision. If we let their velocities be v1 and v2:

$$Mu = Mv_1 + mv_2$$

At this point, we would need to know one of the velocities afterwards in order to calculate the other.

Alternatively, we could have one ball of mass M colliding with another ball of mass m, with both balls moving in opposite directions with velocities u1 and u2 respectively. If we define the direction of motion of the ball with mass M as the positive direction:

$$Mu_1 - mu_2 = Mv_1 + mv_2$$

We do not need to worry about the signs on the right-hand side: they will take care of themselves. If one of our velocities turns out to be negative, we know that it is in the opposite direction to u1.

Elasticity


Although momentum within a closed system is always conserved, kinetic energy does not have to be. If kinetic energy is conserved in a collision, then it is known as a perfectly (or totally) elastic collision. If it is not conserved, then the collision is inelastic. If the colliding particles stick together, then a totally inelastic collision has occurred. This does not necessarily mean that the particles have stopped. In a totally inelastic collision, the two particles become one, giving the equation:

$$Mu_1 + mu_2 = (M + m)v$$

$$v = \frac{Mu_1 + mu_2}{M + m}$$

Explosions
In an explosion, two particles which are stuck together are no longer stuck together, and so gain separate velocities:

$$(M + m)u = Mv_1 + mv_2$$