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?
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