Principles of Physics: A Calculus-Based Text
Principles of Physics: A Calculus-Based Text
5th Edition
ISBN: 9781133104261
Author: Raymond A. Serway, John W. Jewett
Publisher: Cengage Learning
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Chapter 3, Problem 2P

(a)

To determine

The average velocity during time interval t=2.00 s to t=4.00 s .

(a)

Expert Solution
Check Mark

Answer to Problem 2P

The average velocity during time interval t=2.00 s to t=4.00 s is (1.00 m)i^+(0.75 m)j^ .

Explanation of Solution

Section 1:

To determine: The position vector ri .

Answer: The position vector ri is (3.00 m)i^+(1.5 m)j^ .

Given information:

The value of a is m/s , the value of b is 1.00 m , the value of c is 0.125 m/s2 and the value of d is 1.00 m .

The position vector at time t=2.0s is written as,

ri=x(t)i^+y(t)j^=(at+b)i^+(ct2+d)j^

Substitute 2.00 s for t , m/s for a , 1.00 m for b , 0.125 m/s2 for c and 1.00 m for d in above equation to find ri .

ri=((m/s)(2.00 s)+(1.00 m))i^+((0.125 m/s2)(2.00 s)2+(1.00 m))j^=(3.00 m)i^+(1.5 m)j^

Section 2:

To determine: The position vector rf .

Answer: The position vector rf is (5.00 m)i^+(3.00 m)j^ .

Given information:

The value of a is m/s , the value of b is 1.00 m , the value of c is 0.125 m/s2 and the value of d is 1.00 m .

The position vector at time t=4.0s is written as,

rf=(at+b)i^+(ct2+d)j^

Substitute 4.00 s for t , m/s for a , 1.00 m for b , 0.125 m/s2 for c and 1.00 m for d in above equation to find ri .

rf=((m/s)(4.00 s)+(1.00 m))i^+((0.125 m/s2)(4.00 s)2+(1.00 m))j^=(5.00 m)i^+(3.00 m)j^

Section 3:

To determine: The average velocity during time interval t=2.00 s to t=4.00 s .

Answer: The average velocity during time interval t=2.00 s to t=4.00 s is (1.00 m)i^+(0.75 m)j^ .

Given information:

The value of a is m/s , the value of b is 1.00 m , the value of c is 0.125 m/s2 and the value of d is 1.00 m .

The formula to calculate average velocity is,

vavg=rfritfti

Substitute (3.00 m)i^+(1.5 m)j^ for ri , (5.00 m)i^+(3.00 m)j^ for rf , 2.00 s for ti and 4.00 s for tf in above equation to find vavg .

vavg=((5.00 m)i^+(3.00 m)j^)((3.00 m)i^+(1.5 m)j^)(4.00 s2.00 s)=(5.00m3.00m)i^+(3.00m1.5m)j^=(2.00 m)i^+(1.5 m)j^(2.00 s)=(1.00 m)i^+(0.75 m)j^

Conclusion:

Therefore, the average velocity during time interval t=2.00 s to t=4.00 s is (1.00 m)i^+(0.75 m)j^ .

(b)

To determine

The velocity and speed at time t=2.00 s .

(b)

Expert Solution
Check Mark

Answer to Problem 2P

The velocity at time t=2.00 s is (m/s)i^+(0.5 m/s)j^ and the speed at time t=2.00 s is 1.11 m/s .

Explanation of Solution

Section 1:

To determine: The velocity at time t=2.00 s .

Answer: The velocity at time t=2.00 s is (m/s)i^+(0.5 m/s)j^ .

Given information:

The value of a is m/s , the value of b is 1.00 m , the value of c is 0.125 m/s2 and the value of d is 1.00 m .

The formula to calculate velocity at time t=2.00 s is,

v=limΔt0ΔrΔt=drdt=d(((at+b)i^+(ct2+d)j^))dt=ai^+(2ct)j^

Substitute 2.00 s for t , m/s for a and 0.125 m/s2 for c in above equation to find v .

v=(m/s)i^+(2(0.125 m/s2)(2.00 s))j^=(m/s)i^+(0.5 m/s)j^

Conclusion:

Therefore, the velocity at time t=2.00 s is (m/s)i^+(0.5 m/s)j^ .

Section 2:

To determine: The speed at time t=2.00 s .

Answer: The speed at time t=2.00 s is 1.12 m/s .

Given information:

The value of a is m/s , the value of b is 1.00 m , the value of c is 0.125 m/s2 and the value of d is 1.00 m .

The formula to calculate speed at t=2.00 s is,

v=|v|

Calculate the magnitude of velocity as,

|v|=(m/s)2+(0.5 m/s)2=1.12 m/s

Conclusion:

Therefore, the speed during time interval t=2.00 s is 1.12 m/s .

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Chapter 3 Solutions

Principles of Physics: A Calculus-Based Text

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