Question

A body falls freely on to a hard horizontal surface. If the coefficient of restitution between the surface and the body is \( 0.8 \), then the ratio of the maximum height to which the body rises after second impact and the initial height of the body is

• A. \( 256 : 625 \)
• B. \( 64 : 125 \)
• C. \( 16 : 25 \)
• D. \( 4 : 5 \)

Hint:

The height after each bounce is proportional to the square of the coefficient of restitution. After \( n \) impacts, height is \( h_n = e^{2n} h \).

Answer 1

The Correct Option is A

Step 1: Let initial height be \( h \). The body rebounds to a height \( h_1 = e^2 h \) after the first impact, and then again to \( h_2 = e^2 h_1 = e^4 h \) after the second impact. Step 2: Coefficient of restitution \( e = 0.8 \) \[ \Rightarrow \text{Ratio} = \frac{h_2}{h} = \frac{e^4 h}{h} = e^4 = (0.8)^4 \] Step 3: Calculate \( (0.8)^4 \): \[ (0.8)^4 = \left(\frac{8}{10}\right)^4 = \left(\frac{4}{5}\right)^4 = \frac{256}{625} \] % Final Answer \[ \boxed{\frac{256}{625}} \]