Question #1988

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 by the function below. If the raindrop is initially 500 m above the ground, how long does it take to fall? (Give your answer correct to two decimal places.)

<br>a={(9-0.9t ( if ) 0<=t<=10,

0 ( if ) t > 10)

<br><br>____ s

<br>a={(9-0.9t ( if ) 0<=t<=10,

0 ( if ) t > 10)

<br><br>____ s

Expert's answer

Let's find what distance the drop would pass in 10 seconds.

v(t<10) = Int(a(t) dt) +v(0) = 9t - 0.45t^{2} + 10

v(t=10) = 90 - 45 + 10 = 55 m/s

s(t<10) = Int(v(t) dt) = 4.5 t^{2} - 0.15t^{3} + 10t

s(t=10) = 450 - 150 + 100 = 400 m

The rain drop will move 500-400 = 100 m with the constant speed of 55 m/s.

t_{2} = 100/55 = 1.82 s

Thus the total time of the motion is 10 + 1.82 = 11.82 s.

v(t<10) = Int(a(t) dt) +v(0) = 9t - 0.45t

v(t=10) = 90 - 45 + 10 = 55 m/s

s(t<10) = Int(v(t) dt) = 4.5 t

s(t=10) = 450 - 150 + 100 = 400 m

The rain drop will move 500-400 = 100 m with the constant speed of 55 m/s.

t

Thus the total time of the motion is 10 + 1.82 = 11.82 s.

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