Question:

A small ball of mass 0.2 kg is initially at rest on a frictionless surface. It is struck by a larger ball of mass 0.5 kg moving with a velocity of 5 m/s. After the collision, the larger ball moves forward with a velocity of 3 m/s. Determine the final velocity of the smaller ball.

(Assume the collision is perfectly elastic and consider the motion to be in one dimension.)

Answer:

In this collision problem, we can apply the conservation of angular momentum to find the final velocity of the smaller ball.

The angular momentum L of an object is given by the equation: L = Iω, where I is the moment of inertia and ω is the angular velocity.

For a point mass moving in a circular path, the moment of inertia I is equal to the product of the mass and the square of the distance from the axis of rotation: I = mr^2.

Since the surface is frictionless, the angular momentum is conserved before and after the collision.

Before the collision, the larger ball is moving towards the smaller ball, and the smaller ball is initially at rest. Thus, the initial angular momentum (L_initial) is zero.

After the collision, the larger ball moves forward with a velocity of 3 m/s, and the smaller ball moves with a final velocity v.

Let's use the conservation of angular momentum equation to solve for v:

L_initial = L_final

(0.5 kg)(5 m/s)(r) = (0.5 kg)(3 m/s)(r') + (0.2 kg)(v)(r')

Simplifying the equation:

2.5r = 1.5r' + 0.2vr'

Now, we need to find the unknowns r and r'. We know that r' is the distance of the smaller ball from the axis of rotation after the collision, and r is the distance of the larger ball from the axis of rotation before the collision.

Since the situation is one-dimensional, we can assume r' = -r, as the smaller ball moves in the opposite direction compared to the initial motion of the larger ball.

Substituting r' = -r into the equation:

2.5r = 1.5(-r) + 0.2vr(-1)

2.5r = -1.5r - 0.2vr

Adding 1.5r + 0.2vr to both sides:

2.5r + 1.5r + 0.2vr = 0

4r + 0.2vr = 0

Factoring out r:

r(4 + 0.2v) = 0

Since r cannot be zero (as it represents the distance from the axis of rotation), we can set the term in parentheses to zero:

4 + 0.2v = 0

0.2v = -4

v = -20 m/s

The final velocity of the smaller ball is -20 m/s, indicating that it is moving in the opposite direction to the initial motion of the larger ball.

Thus, the final velocity of the smaller ball is -20 m/s.