# The equation [ v ( t ) ] 2 = v 0 2 − 19.6 [ s ( t ) − s 0 ] . ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805 ### Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

#### Solutions

Chapter 4.8, Problem 45E
To determine

## To show: The equation [v(t)]2=v02−19.6[s(t)−s0] .

Expert Solution

The proof of statement [v(t)]2=v0219.6[s(t)s0] for object that is projected upward is shown.

### Explanation of Solution

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)=4.9t2+v0t+D (5)

Here,

D is arbitrary constant.

Substitute 0 for t in equation (5),

s(0)=4.9(02)+v0(0)+D=0+D=D

Substitute s0 for s(0) ,

D=s0

Substitute s0 for D in equation (5),

s(t)=4.9t2+v0t+s0

Apply square on both sides of equation (4).

[v(t)]2=(9.8t+v0)2=(v0)2+(9.8t)22(9.8t)v0=v02+96.04t219.6v0

Add and subtract the term s0 on RHS of expression.

[v(t)]2=v02+96.04t219.6v0s0+s0=v0219.6[(4.9t2+v0+s0)s0]

Substitute s(t) for (4.9t2+v0+s0) ,

[v(t)]2=v0219.6[s(t)s0]

Thus, the proof of statement [v(t)]2=v0219.6[s(t)s0] for object that is projected upward is shown.

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