Question

A steel ball of mass $m=50g$ falls from the height $h=1.0m$ on the horizontal surface of a massive slab. Find the cumulative momentum that the ball imparts to the slab after numerous bounces (in $gm/s$ ), if every impact decreases the velocity of the ball $\eta =1.25$ times.

Answer 1

$Impulseimpartedafter1stcollisionismu(1+\frac{1}{\eta })Impulseimpartedafter2ndcollisionism\frac{u}{\eta }(1+\frac{1}{\eta })Impulseimpartedafter{i}^{th}collisionism\frac{u}{{\eta }^{i-1}}(1+\frac{1}{\eta })totalImpulseimparted=mu(1+\frac{1}{\eta })(1+\frac{1}{\eta }+\frac{1}{{\eta }^2}+....)=mu(1+\frac{1}{\eta })\frac{\eta }{\eta -1}=.05\times \sqrt{2\times 10}\times (1+\frac{4}{5})\left(\frac{5/4}{5/4-1}\right)=2.01$