A ball dropped from a height of \(4 \mathrm{~m}\) rebounds to a height of $2.4 \mathrm{~m}$ after hitting the ground. Then the percentage of energy lost is (A) 40 (B) 50 (C) 30 (D) 600

To find the initial potential energy (PE_initial) and final potential energy (PE_final), we will use the formula for potential energy: \[PE = mgh\] where \(m\) = mass of the ball (unknown in this case, but it will cancel out in the end), \(g\) = acceleration due to gravity (\(9.81 m/s^2\)), \(h\) = height of the ball. PE_initial will be when the ball is dropped from a height of 4m while PE_final will be when the ball rebounds to a height of 2.4m.

To calculate the energy loss, we will subtract the final potential energy (PE_final) from the initial potential energy (PE_initial): \[Energy\,Loss = PE_{initial} - PE_{final}\]

To find the percentage of energy lost, we will divide the energy loss by the initial potential energy and then multiply it by 100: \[%Energy\,Lost = \frac{Energy\,Loss}{PE_{initial}} \times 100\] Now let's plug in the values and calculate the percentage of energy lost:

Using the potential energy formula, we get: \( \begin{cases} PE_{initial} = m \times 9.81 \times 4 \\ PE_{final} = m \times 9.81 \times 2.4 \end{cases} \) Therefore, the energy loss is: \[Energy\,Loss = (m \times 9.81 \times 4 ) - (m \times 9.81 \times 2.4)\] Now we will calculate the percentage of energy lost: \[%Energy\,Lost = \frac{(m \times 9.81 \times 4 ) - (m \times 9.81 \times 2.4)}{m \times 9.81 \times 4} \times 100\] Simplifying the equation, we get: \[%Energy\,Lost = \frac{9.81m (4 - 2.4)}{9.81m \times 4} \times 100\] Notice that the mass "m" and the acceleration due to gravity "9.81" cancel out: \[%Energy\,Lost = \frac{(4 - 2.4)}{4} \times 100\] Thus, \[%Energy\,Lost = (1 - 0.6) \times 100 = 0.4 \times 100\] \[%Energy\,Lost = 40\%\] So, the percentage of energy lost for the ball is 40% which corresponds to option (A).