# The velocity of the particle when t = a , t = 1 , t = 2 and t = 3 seconds.

### 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 2.6, Problem 15E
To determine

## To find: The velocity of the particle when t=a, t=1, t=2 and t=3 seconds.

Expert Solution

The velocity of the particle at times t=a,t=1,t=2 and t=3 are 2a3 m/s_, 2 m/s_ , 14 m/s_ and 227 m/s_, respectively.

### Explanation of Solution

Given:

The displacement (in meters) of a particle moving in a straight line is s=1t2, where t is measured in seconds.

Formula used:

The derivative v(a) is the velocity of the particle at time t=a,

v(a)=limh0f(a+h)f(a)h (1)

Calculation:

Obtain the velocity of the particle at t=a.

Take the position function s(t)=f(t) and use equation (1) to compute the velocity of the particle at time t=a as follows.

v(a)=limh0f(a+h)f(a)h=limh0(1(a+h)2)(1(a)2)h=limh0(1(a2+h2+2ah))(1a2)h=limh0(a2(a2+h2+2ah)a2(a2+h2+2ah))h

Simplify further and obtain the velocity as follows.

v(a)=limh0(h2+2ah)ha2(a2+h2+2ah)=limh0h(h+2a)ha2(a2+h2+2ah)

Cancel the common term h from both the numerator and the denominator,

v(a)=limh0(h+2a)a2(a2+h2+2ah)=(0+2a)a2(a2+(0)2+2a(0))=2aa2(a2)=2a3

Thus, the velocity of the particle at time t=a is 2a3 m/s_.

Obtain the velocity of the particle at time t=1 second.

Substitute a=1 s in v(a),

v(1)=2(1)3=2

Thus, the velocity of the particle at time t=1 s is 2 m/s_.

Obtain the velocity of the particle at time t=2 seconds.

Substitute a=2 s in v(a),

v(2)=2(2)3=28=14

Thus, the velocity of the particle at time t=2 s is 14 m/s_ .

Obtain the velocity of the particle at time t=3 seconds.

Substitute a=3 s in v(a),

v(3)=2(3)3=227

Thus, the velocity of the particle at time t=3 s is 227 m/s_.

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