Problem 7.

Ball A, with a mass of 2 kg, moves with a velocity 5 m/s. It collides with a stationary ball B, with a mass of 4 kg. After the collision, ball A moves in a direction 60.0 degrees to the left of its original direction, while ball B moves in a direction 50.0 degrees to the right of ball A's original direction. Calculate the velocities of each ball after the collision.

Solution:

In the present problem we need to use the momentum conservation law: the net momentum of two balls before the collision is equal to the net momentum of two balls after the collision. It is important that momentum is a vector. Then the net momentum of two balls is a vector sum of the momentum (vector) of ball A and the momentum (vector) of ball B.

Therefore, the initial net momentum of two balls is

A momentum of a ball is a product of a mass of a ball and its velocity. Then

The final momentum is

Then the momentum conservation law takes the form

or

.......................(1)

To solve this vector equation we need to introduce coordinate system (axis x and axis y) as shown in the figure.

We follow the standard procedure of finding the components of vectors and rewrite the vector equation (1) in terms of x and y-components:

x-component of equation (1):

y-component of equation (1):

Finally, we have system of two equations with two unknown variables

From the second equation we obtain

We substitute this expression in the first equation:

Then

Therefore the velocities of the balls after collision are 22.9 m/s and 12.8 m/s (the initial kinetic energy is less than the final energy, which means that during the collision there is an additional source of energy).