A ball is dropped from a height of 20m. If the coefficient of restitution for the collision between ball and floor is 0.5, after hitting the floor, the ball rebounds to a height of ________ m.

JEE Mains - Ball Rebound Height

Question: A ball is dropped from a height of \(20\,\text{m}\). If the coefficient of restitution for the collision between the ball and the floor is \(0.5\), after hitting the floor, the ball rebounds to a height of _______ m.
[31-Jan-2023 Shift 1]

Select the correct answer:

(A) \(2.5\,\text{m}\)

(B) \(5\,\text{m}\)

(C) \(10\,\text{m}\)

(D) \(20\,\text{m}\)

Solution

When the ball is dropped from \(20\,\text{m}\), its speed just before impact is given by: \[ v = \sqrt{2gh} = \sqrt{2 \times 10 \times 20} = \sqrt{400} = 20\,\text{m/s}. \]

The coefficient of restitution \(e\) is defined as: \[ e = \frac{\text{rebound speed}}{\text{incident speed}}. \] Thus, the rebound speed is: \[ v_{\text{rebound}} = e \times 20 = 0.5 \times 20 = 10\,\text{m/s}. \]

The maximum height reached after rebound can be found using: \[ v_{\text{rebound}}^2 = 2gh' \quad \Longrightarrow \quad h' = \frac{v_{\text{rebound}}^2}{2g} = \frac{10^2}{2 \times 10} = \frac{100}{20} = 5\,\text{m}. \]

Therefore, the ball rebounds to a height of \(5\,\text{m}\).