# How long does the raindrop take to fall to the ground? Since raindrops grow as they fall, their surface area increases and therefore the resistance to their falling increases. A raindrop has an...

How long does the raindrop take to fall to the ground?

Since raindrops grow as they fall, their surface area increases and therefore the resistance to their falling increases. A raindrop has an initial downward velocity of 10 m/s and its downward acceleration is given

`a = 3 -0.3t`   if `0<=t<=10`
`a = 0` if `t>10`

If the raindrop is initially 500 m above the ground, how long does it take to fall?

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The velocity in the time up to 10 seconds is given by

`10 + int (3-0.3t)dt = 3t - 0.15t^2 + 10`

The distance travelled in this time is

`int_0^10 (3t-0.15t^2 +10)dt = (3/2t^2 -0.05t^3 + 10t)|_0^10 = `

`3/2(100) - 0.05(1000) +100` `= 150 - 50 + 100 = 200` m

The raindrop at that point has another 300m to fall

The velocity after 10s = 30 - 0.15(100) + 10 = 25m/s

It remains at this speed when t>10 as it has stopped accelerating

To fall 300m at this speed it takes 300/25 = 12s

Therefore the raindrop takes 10 + 12 = 22s to fall to the ground

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