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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Vertical Motion In Exercises 65–68, use s " ( t ) = 32 feet per second per second as the acceleration due to gravity. (Neglect air resistance.) See Example 7.

The Grand Canyon is 6000 feet deep at the deepest point. A rock is dropped from the rim above this point. Express the height s (in feet) of the rock as a function of the time t (in seconds). How long will it take the rock to hit the canyon floor?

To determine

To calculate: The height s (in feet) of the rock as a function of time which is thrown from the rim of a 6000 feet Grand Canyon to deepest point and also the time takes by the rock to hit the canyon floor.

Explanation

Given Information:

The depth of the deepest point from the rim of the Grand Canyon is 6000 feet and the acceleration due to gravity in downward direction is 32feet per second per second.

Formula used:

The constant rule of integration, kdx=kx+C.

The simple power rule of integration, xndx=xnn+1+C.

Calculation:

Consider the height of the rock at t time is s.

The acceleration function due to gravity in downward direction is 32feet per second per second.

s''(t)=32

Integrate both sides, use the constant rule of integration kdx=kx+C.

s''(t)dt=32dts'(t)=32t+C1

Here, s'(t) is the velocity of the rock at time t.

Substitute 0 for t in the velocity function s'(t)=32t+C1.

s'(0)=32(0)+C1=C1

The rock is drop from the rim so its initial velocity is zero.

s'(t)=0

Substitute 0 for t in the velocity function s'(t)=32t+C1.

s'(0)=32(0)+C1=C1

Substitute 0 for s'(0) in above function for initial velocity condition.

0=0+C1C1=0

Therefore, the value of constant C1 is 0.

Substitute 0 for C1 in velocity function s'(t)=32t+C1

s'(t)=32t+0=32t

Integrate both side the above velocity function, use the simple power rule of integration xndx=xnn+1+C

s'(t)dt=32tdts(t)=32(t1+11+1)+C2=32(t22)+C2=16t2+C2

Substitute 0 for t in above height function for initial condition.

s(0)=16(0)2+C2s(0)=C2

Substitute 6000 for s(0) in above height function for initial height condition

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