A rubber ball is dropped from a height of \(5 \mathrm{~m}\) on a planet, where the acceleration due to gravity is not known. On bouncing it rises to \(1.8 \mathrm{~m}\). The ball loses its velocity on bouncing by a factor of

Key Idea According to conservation of energy, potential energy at height \(h=\) kinetic energy at
ground. Potential energy = Kinetic energy
ie,
\(m g h=\frac{1}{2} m v^{2}\)
\(\Rightarrow \quad v=\sqrt{2 g h}\)
If \(h_{1}\) and \(h_{2}\) are initial and final heights, then
\(v_{1}=\sqrt{2 g h_{1}}, \quad v_{2}=\sqrt{2 g h_{2}}\)
Loss in velocity
\(\Delta v=v_{1}-v_{2}=\sqrt{2 g h_{1}}-\sqrt{2 g h_{2}}\)
\(\therefore\) Fractional loss in velocity \(=\frac{\Delta v}{v_{1}}\)
\(\begin{array}{l}=\frac{\sqrt{2 g h_{1}}-\sqrt{2 g h_{2}}}{\sqrt{2 g h_{1}}} \\=1-\sqrt{\frac{h_{2}}{h_{1}}}\end{array}\)
Substituting the values, we have
\(\therefore\)
\(\frac{\Delta v}{v_{1}}=1-\sqrt{\frac{1.8}{5}}\)
\(\begin{array}{l}=1-\sqrt{0.36}=1-0.6 \\=0.4=\frac{2}{5}\end{array}\)