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A ball is dropped from a height of 1.8 m onto a horizontal floor and rebounds with coefficient of restitution e=0.5 each time. Treating each…

Numerical answer

A ball is dropped from a height of

1.8\ \mathrm{m}

onto a horizontal floor and rebounds with coefficient of restitution

e=0.5

each time. Treating each collision as instantaneous, the total distance travelled by the ball (in cm) before it comes to rest is

Show answer and solution

Answer: 300

After the first fall of

h

, rebound heights are

e^2h,\ e^4h,\dots

and each rebound height is traversed up and down. Total distance

=h+2h(e^2+e^4+\cdots)=h+\dfrac{2he^2}{1-e^2}=h\,\dfrac{1+e^2}{1-e^2}

. With

e^2=0.25

1.8\times\dfrac{1.25}{0.75}=1.8\times\dfrac53=3.0\ \mathrm{m}=300\ \mathrm{cm}

. Using

e

instead of

e^2

for heights is the usual error.

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