Question

A ball falls from a height \( h \) upon a fixed horizontal floor. The coefficient of restitution is \( e \). The total distance covered by the ball before coming to rest (neglect air resistance) is

• A. \( \frac{1-e^2}{1+e^2}h \)
• B. \( \frac{1+e^2}{1-e^2}h \)
• C. \( \frac{1-2e^2}{1+e^2}h \)
• D. \( \frac{1+2e^2}{1-e^2}h \)

Hint:

Restitution problems: \begin{itemize} \item Height ratio after bounce = \( e^2 \). \item Use geometric series. \end{itemize}

Answer 1

The Correct Option is B

Concept: Each bounce height reduces by factor \( e^2 \). Step 1: {\color{red}Distances travelled.} Initial fall: \[ h \] Upward and downward after first bounce: \[ 2eh,\; 2e^2h,\; 2e^3h,\dots \] Actually heights scale by \( e^2 \), so distances: \[ 2he^2,\; 2he^4,\dots \] Step 2: {\color{red}Total distance.} \[ S = h + 2h(e^2 + e^4 + e^6 + \dots) \] Step 3: {\color{red}Geometric series.} \[ \sum e^{2n} = \frac{e^2}{1-e^2} \] \[ S = h + 2h \frac{e^2}{1-e^2} \] Step 4: {\color{red}Simplify.} \[ S = \frac{h(1+e^2)}{1-e^2} \]