Question:

A 0.5 kg cart is initially at rest on a frictionless track. A 0.1 kg ball is thrown horizontally towards the cart with a velocity of 10 m/s. As the ball collides with the cart, it experiences an elastic collision and rebounds with a velocity of 8 m/s in the opposite direction. What is the final velocity of the cart after the collision? Provide your answer in m/s.

Answer:

To determine the final velocity of the cart after the collision, we can use the laws of momentum and conservation of momentum.

We know that the momentum of an object is given by the product of its mass (m) and its velocity (v), represented by the equation:

p = m * v

According to the conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision.

Initially, the cart is at rest, so its momentum is zero. The momentum of the ball before the collision can be calculated as:

momentum_ball_before = (mass_ball) * (velocity_ball)
                     = (0.1 kg) * (10 m/s)
                     = 1 kg⋅m/s

During the collision, the ball rebounds in the opposite direction, so its final momentum is:

momentum_ball_after = -(mass_ball) * (velocity_ball)
                    = -(0.1 kg) * (8 m/s)
                    = -0.8 kg⋅m/s

According to the conservation of momentum, the final momentum of the system (cart + ball) after the collision should equal zero since the cart was initially at rest:

momentum_cart_after + momentum_ball_after = 0
momentum_cart_after = -momentum_ball_after
                  = -(-0.8 kg⋅m/s)
                  = 0.8 kg⋅m/s

We know that the momentum of an object is related to its velocity, so:

momentum_cart_after = (mass_cart) * (velocity_cart_after)
0.8 kg⋅m/s = (0.5 kg) * (velocity_cart_after)

Solving for velocity_cart_after:

velocity_cart_after = 0.8 kg⋅m/s / 0.5 kg
velocity_cart_after = 1.6 m/s

Therefore, the final velocity of the cart after the collision is 1.6 m/s.