Question

A certain ball when dropped to the ground rebounds to 4/5th of the total height from which it falls; it is dropped from a height of 100 metres. Find the total distance it travels before coming to the rest.​

Answer 1

Answer:

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Step-by-step explanation:

First drop is 100m

$\frac{4}{5} th \ of \ 100=\frac{100*4}{5}=80m$

This process will continu in the form of infinte G.P.

So, common ratio $=\frac{80}{100}=0.8\\ \\\frac{a}{1-r} \\ \[(\frac{100}{\frac{1}{5} }+\frac{80}{\frac{1}{5} } )]\\\\=500+400=900m$

Answer 2

Step-by-step explanation:

Answer :

Given Bouncing ball rebound each time ti a height one half of the height of previous bounce ,

And it Dropped from the height of 10 m , So

1 ) Total distance travelled when it hits the ground 10th time .

So ,

we add all height the ball going to achieve until it hits 10th time .

SO,

10m now first bounce + 102 Bounce again and + 102 Drop to bounce second time + 104 Bounce again and + 104 Drop to bounce third time + ...

So we get

10 + 102 +102 + 104 + 104 + 108 + 108 + 1016 + 1016 + 1032 + 1032 + 1064 + 1064 + 10128 + 10128 + 10256 + 10256 + 10512 + 105122nd Bounce 3rd 4th 5th 6th 7th 8th 9th 10th

10 + 10 + 102 +104 + 108 + 1016 + 1032 + 1064 + 10128 + 10256

20 + 10 [ 12 +14 + 18 + 116 + 132 + 164 + 1128 + 1256 ]

Now we can see that inside bracket we have a GP , where r ( Common ratio ) = 12 And a ( first term ) = 12 And n ( number of terms = 8

And We know Sum of GP = a (1 − rn1 − r)

SO we get Sum of our GP = 12 (1 − (12)91 − 12) = 12(1 − 12561 − 12) =12 (256 − 1256 2 − 12) = 12(25525612) = 255256 = 0.99609375

SO we get

= 20 + 10 ( 0.99609375)

= 20 + 9.9609375

= 29.9609375

So,

Total distance covered by ball until it hits 10th time on the ground = 29.9609375 m ( Ans )

2 )According to given condition ball will bounce infinitely times as it bounce half of its previous height every time . We get AS :

20 + 10 [ 12 +14 + 18 + 116 + 132 + 164 + 1128 + 1256 + ... ]

And we know Formula for Sum of infinitely terms is As :

Sum of infinitely GP = a(11 − r)

So, we get

Sum of infinitely sum = 12(11 − 12) = 12(12 − 12) =12 (112) = 1

So we get

20 + 10 ( 1 ) = 20 + 10 = 30

SO we can say the total distance it travels before coming to rest. = 30 m ( Ans )