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Question

After striking the floor, a certain ball rebounds $${ (4/5) }^{ th }$$ of height from which it has fallen. Then the total distance that it travels before coming to rest, if it is gently dropped from a height of 120 m is 


A
1260 m
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B
600 m
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C
1080 m
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D
none of these
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Solution

The correct option is C 1080 m
Initially, the ball falls from a height of $$120\ m$$. 
After striking the floor, it rebounds and goes to a height of $$\dfrac{4}{5} \times (120)\ m $$. 
Now, it falls from a height  of $$\dfrac{4}{5} \times (120)\ m $$ and after rebounding goes to a height of $$\dfrac{4}{5} \left(  \dfrac{4}{5} (120) \right) m$$. 

This process is continued till the ball comes to rest.

Hence, the total distance travelled is
$$D= 120 + 2 \left[ \dfrac{4}{5} (120) +\left( \dfrac{4}{5} \right)^2 (120) + \dots \infty \right]$$
The above series in brackets is an infinte GP,
$$\therefore D = 120 + 2 \left[ \dfrac{\dfrac{4}{5} (120)}{1 - \dfrac{4}{5}} \right] \\ \quad= 120+2\left[\dfrac{480}{1}\right]= 1080 \ m$$

Mathematics

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