Answer to Question #1988 in Calculus for Abhilash

Question

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.)


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




____ s

Answer 1

Let's find what distance the drop would pass in 10 seconds.
v(t<10) = Int(a(t) dt) +v(0) = 9t - 0.45t² + 10
v(t=10) = 90 - 45 + 10 = 55 m/s
s(t<10) = Int(v(t) dt) = 4.5 t² - 0.15t³ + 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.
t₂ = 100/55 = 1.82 s
Thus the total time of the motion is 10 + 1.82 = 11.82 s.