Student Question

If a stone is dropped from a balloon rising at the rate of 20 fps reaches the ground in 10 seconds, find the initial height and final velocity.

Expert Answers

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The equation for a free falling body (no acceleration except gravity -- thus ignoring air resistance, etc...) is

`h(t)=-16t^2+v_0t+h_0` where

h -- the height in feet at time t

t -- the time in seconds

`v_0` -- the initial velocity in feet per second.. (Positive if away from the Earth, and negative if towards the Earth.)

`h_0` -- the initial height in feet.

The velocity function is the derivative of the displacement function so with v as the velocity in feet per second at time t we get:

`v(t)=-32t+v_0`

To find the initial height, we use the given information that at time t=10 the height is 0 and the initial velocity is 10fps (again positive since it is directed away from the ground):

`0=-16(10)^2+10(10)+h_0`

`=>h_0=1500"ft"`

To find the final velocity we substitute t=10 into the velocity formula:

`v_(10)=-32(10)+10` or v=-310fps.

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The initial height was 1500 ft and the final velocity was -310 fps.

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