anonymous
  • anonymous
Haziq drops a ball from a height of H cm above the floor.After the 1st bounce,the ball reaches a height of h_1 cm where H_1=2(H)/3.After the 2nd bounce,the ball reaches a height of H_2 cm where H_2=2(H_1)/3.The ball continue bouncing in this way until it stops. Given that H=600cm,find b) the total distance, in cm,travelled by the ball until it stops. @ganeshie8
Mathematics
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SOLVED
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schrodinger
  • schrodinger
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ganeshie8
  • ganeshie8
H = 600cm = 6m Plug that in and can you list the first few terms(heights of bounces) ?
anonymous
  • anonymous
\[600,400,266\frac{ 2 }{ 3 },177\frac{ 7 }{ 9 },118\frac{ 14 }{ 27 },...\]
ganeshie8
  • ganeshie8
Do not simplify, leave 600 as it is in the terms

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More answers

ganeshie8
  • ganeshie8
\(H_0=600\) \(H_1=600(2/3)\) \(H_2=600(2/3)^2\) \(H_3=600(2/3)^3\) \(\vdots\)
ganeshie8
  • ganeshie8
Those are the heights at each step, yes ?
anonymous
  • anonymous
yes
ganeshie8
  • ganeshie8
Now, look at what the ball is doing
ganeshie8
  • ganeshie8
It is falling freely from a height of 600cm, so it must travel a distance of 600cm before hitting the ground for the first time, yes ?
anonymous
  • anonymous
yes
ganeshie8
  • ganeshie8
|dw:1449668961684:dw|
ganeshie8
  • ganeshie8
After that, it reaches a height of \(H_1\) and also it falls by the same distance
ganeshie8
  • ganeshie8
If you see, except for \(H_0\), all other heights are travelled "TWO" times. One while rising, other while falling.
ganeshie8
  • ganeshie8
remember the infinite geometric series formula ?
anonymous
  • anonymous
yes
ganeshie8
  • ganeshie8
The distances travelled by the ball looks something like below : \(D_0=600\) \(D_1=\color{red}{2}*600(2/3)\) \(D_2=\color{red}{2}*600(2/3)^2\) \(D_3=\color{red}{2}*600(2/3)^3\) \(\vdots\)
ganeshie8
  • ganeshie8
Forget \(D_0\). Observe that the other terms form a geometric progression with \(r = 2/3\)
ganeshie8
  • ganeshie8
\(D_1=\color{red}{2}*600(2/3)\) \(D_2=\color{red}{2}*600(2/3)^2\) \(D_3=\color{red}{2}*600(2/3)^3\) \(\vdots\)
ganeshie8
  • ganeshie8
first term = \(\color{red}{2}*600(2/3)\) common ratio = \(2/3\) can you find the infinite sum using the geometric series formula ?
anonymous
  • anonymous
\[S_\infty=\frac{ a }{ 1-r }\]\[=\frac{ 800 }{ 1-\frac{ 2 }{ 3 } }\]
ganeshie8
  • ganeshie8
Yes, simplify
anonymous
  • anonymous
\[=2400\]
ganeshie8
  • ganeshie8
Looks good. Add the initial height \(H_0\) to that...
ganeshie8
  • ganeshie8
|dw:1449669495868:dw|
anonymous
  • anonymous
\[Total~distance=600+2400\]\[=3000\]
ganeshie8
  • ganeshie8
Looks good !
ganeshie8
  • ganeshie8
total distance = 3000 cm
ganeshie8
  • ganeshie8
maybe express it as meters
anonymous
  • anonymous
Thank you @ganeshie8 :)
ganeshie8
  • ganeshie8
np :)

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