## Solution using conservation of momentum and energy

before the collision the two bodies are approaching in the same plane:

after the collision they have new velocities but are still in the same plane:

(note: all velocities are positive toward the right, the arrows on the diagrams above just indicate their motion relative to each other)

by conservation of energy: * + * = * + * (both sides of equation multiplied by two)

by conservation of momentum: * + * = * + *

where:

is initial velocity of body a | is initial velocity of body b |

is final velocity of body a | is final velocity of body b |

is mass of body a | is mass of body b |

We want to find the final velocities ( and ) so there are 2 unknowns and 2 equations, so we can solve:

This is a simultaneous quadratic equation which is difficult to solve, here is my attempt. But there is a trick, that is to measure in the frame of reference of one of the bodies, that way we can show that after the collision the bodies will move apart at the same rate that they approached each other before the collision:

rearranging 1 gives: * ( - ) = * (- )

giving : * ( + )*( - ) = * (+ ) * (- )

and from 2 we get: * ( - ) = * (- )

dividing 3 by this gives: + = +

so : - = - ( - )

i.e. the bodies move away form each other at minus the speed that they were approaching.

But this only tells you what the relative speed of the objects is to calculate the absolute speed, try the following:

from 2 : * + * = * + *

* = * + * - *

substituting for in 4 gives: ( + - )* = * + * - *

* (+) = * (- )+ *2*

= * + *

and

= + - = + * + *( -1)

= * - *

### perfectly inelastic collision

First take the case of perfectly inelastic collisions (where the objects stick together after collision) and their final velocity is equal.

So,

=

( - )* = -( - )*

* ( + ) = * + *

so the solution is:

= =* + *

The value of the impulse is:

impulse = ( - )*

impulse = (* + * - )*

impulse = (* + * - )*

impulse = (*- + * )*

impulse = (- )*

### perfectly elastic collision

In this case the equation for impulse is the same as inelastic case (but its value is twice because the objects separate at the same rate that they approach)

impulse = ( - )* = - ( - )*

in this case, relative separating velocity = - relative approach velocity

( - ) = -( - )

-( - ) = -( - 2+)

( - )* +( - 2+)* = 0

* - * +( - 2+)* = 0

* ( + )= * ( -) + 2**

which gives the solution -

= * + *

The value of the impulse is:

impulse = ( - )*

impulse = ( * + * - )*

impulse = ( *- + * )*

impulse = (- ) *

impulse = 2(- )*