   Chapter 4.9, Problem 67E ### 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

# An object is projected upward with initial velocity v0, meters per second from a point s0 meters above the ground. Show that [ v ( t ) ] 2 = v 0 2 − 19.6 [ s ( t ) − s 0 ]

To determine

To show: The equation [v(t)]2=v0219.6[s(t)s0] .

Explanation

Given data:

Initial velocity is v0 , and initial displacement is s0 .

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 t is 12t2 and 1 is t .

Calculation:

Initial velocity is v0 , that is v(0)=v0 and the initial displacement is s0 , that is s(0)=s0 .

The motion of stone is close to ground, so the motion is considered as gravitational constant (g) , which is 9.8ms2 .

Write the expression for acceleration function (a(t)) .

a(t)=9.8

Substitute 9.8 for a(t) in equation (1),

v(t)=9.8

Antiderivate the expression with respect to t,

v(t)=9.8t+C (3)

Here,

C is arbitrary constant.

Substitute 0 for t in equations (3),

v(0)=9.8(0)+C=C

Substitute v0 for v(0) ,

C=v0

Substitute v0 for C in equation (3),

v(t)=9.8t+v0 (4)

Substitute 9.8t+v0 for v(t) in equation (2),

s(t)=9.8t+v0

Antiderivate the expression with respect to t,

s(t)=9.8(12t2)+v0t+D

s(t)=4

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