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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

A model rocket is fired vertically upward from rest. Its acceleration for the first three seconds is a ( t ) =   60 t , at which time the fuel is exhausted and it becomes a freely “falling” body. Fourteen seconds later, the rocket’s parachute opens, and the (downward) velocity slows linearly to 18 ft/s in 5 seconds. The rocket then “floats” to the ground at that rate.

(a) Determine the position function s and the velocity function v (for all times t). Sketch the graphs of 5 and v.

(b) At what time does the rocket reach its maximum height, and what is that height?

(c) At what time does the rocket land?

To determine

a)

To determine & sketch:

The position function s and the velocity function v & sketch graphs of s and v.

Explanation

1) Concept:

Theorem: If F is an antiderivative of  f on an interval I, then the most general antiderivative of f on I is Fx+c, where c is an arbitrary constant.

2) Given:

α0t=60t,  0t3

3) Calculations:

Let, α0t=60t,  0t3

Taking the general antiderivative of α0t

v0t=30t2+C, where C  is the constant.

However, t=0, v0t=0

v00=3002+C

So, C=0 then v0t is

v0t=30t2

Now taking the general antiderivative of  v0t

s0t=30t33+D

s0t=10t3+D

However, t=0, s0t=0

s00=10(0)3+D

So, D=0

 s0t=10t3

Now, find out the v0t & s0t at t=3

So, substitute t=3 in v0t & s0t we get,

v03=3032=30*9=270

 s03=1033=10*27=270

For 3t17,

Therefore, the acceleration must be negative then the acceleration due to gravity is

α1t=dv1dt=-32

α1t=-32 ft/s

Taking the general antiderivative of α1t we get,

v1t=-32(t-3)+v0

v1t=-32(t-3)+270 And

Taking the general antiderivative of v1t we get,

s1t=-16t-32+270t-3+s0

s1t=-16t-32+270t-3+270

Now at t=17 substitute in above v1t & s1t we get,

v1t=-178

s1t=914

For 17<t22,

α2t=32

Taking the general antiderivative of α2t we get,

v2t=32

To determine

b)

To find:

The time and the height when the rocket reaches its maximum height.

To determine

c)

To obtain:

The time of landing of the rocket.

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