   Chapter 4.9, Problem 73E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# 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 a = { 9 − 0.9 t if   0 ≤ t ≤ 10 0 if   t > 10 If the raindrop is initially 500 m above the ground, how long does it take to fall?

To determine

The time taken by raindrop to fall.

Explanation

Given data:

Initial downward velocity is 10ms, initial displacement is 500 m.

Consider the downward acceleration as,

a={90.9tif0t100ift>10

Formula Used:

Write the expression for acceleration function a(t).

a(t)=v(t) (1)

Here,

v(t) is first derivative of velocity function v(t).

Write the expression for velocity function v(t).

v(t)=s(t) (2)

Here,

s(t) is first derivative of displacement function s(t).

Antiderivative of tn is tn+1n+1, 0 is K, and 1 is t.

Calculation:

Consider upward direction is indicated by positive sign and downward direction by negative. The initial velocity of raindrop is 10ms downwards, that is v(0)=10 and consider the initial displacement as 500 m, that is s(0)=500.

Write the expression for acceleration (a1(t)) considering time as 0t10.

a1(t)=(90.9t)=0.9t9

Substitute v1(t) for v(t) and 0.9t9 for a(t) in equation (1),

v1(t)=0.9t9

Antiderivate the expression with respect to t.

v1(t)=9t+0.9(12t2)+C

v1(t)=9t+0.45t2+C (3)

Here,

C is arbitrary constant.

Substitute 0 for t in equations (3),

v1(0)=9(0)+0.45(0)2+C=0+C=C

Substitute –10 for v(0),

C=10

Substitute –10 for C in equation (3),

v1(t)=9t+0.45t210 (4)

Substitute s1(t) for s(t) and 9t+0

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