Collisions in two dimensions

Momentum is a vector quantity and as such direction is important in any collision analysis.

Elastic Collisions (during which both Kinetic Energy and Momentum are conserved) can be of two types:

•  HEAD-ON

• NON HEAD-ON

Non Head-on collisions are usually more common.  In sports and games such as billiard and bocce ball these types of collisions are more frequent.  Non head-on collisions also occur in traffic accidents, nuclear experiments and so on.

Example:

During a billiard game, one ball hits a second identical ball at rest, with a speed of 8.2 m/s.
One of the balls goes off at an angle of 60⁰ At what angle will the second ball go off and what will the velocities of the two balls be after the collision?

Solution:

This is an elastic collision.  Therefore, both kinetic energy and momentum are conserved.

Part 1.  Conservation of Kinetic Energy.

Given: m₁ = m₂ (balls are identical);  v₂ = 0 m/s (at rest);  v₁ = 8.2 m/s;  θ = 60⁰

Therefore the equation for the conservation of kinetic energy

 becomes:

....... Equation (1)

Part 2.  Conservation of Momentum.

The equations for the conservation of Momentum

can be written as

   in the X-direction

and     in the y-direction

applying the given conditions  v₂ = 0 m/s, m₁ = m₂, and the fact that there is no initial movement in the y-direction, gives

......... Equation (2)

........... Equation (3)

Form equation 3:

  Equation 4

From Equation 2:

Equation 5

Add  {Equation 4 to Equation 5} and use      

Substitute Equation 1     

v₁ = 8.2 m/s  and   Q₁ = 60⁰

Therefore   v₁^' = (8.2)(cos60⁰) = 4.1 m/s

Use equation 1 to solve for v₂^'

v₂^' = 7.1 m/s

Now find the angle (use equation 2)

Q₂ = 30⁰ 

This result is predictable in view of the fact that the two masses are equal and that the final momentum is a vector perpendicular to the two momentums at any point in time.  Therefore one could have written that Q₂ =  (90⁰- Q₁ ) = (90⁰ - 60⁰) = 30⁰