Physics for Scientists and Engineers with Modern Physics
Physics for Scientists and Engineers with Modern Physics
10th Edition
ISBN: 9781337553292
Author: Raymond A. Serway, John W. Jewett
Publisher: Cengage Learning
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Chapter 38, Problem 1P

In a laboratory frame of reference, an observer notes that Newton’s second law is valid. Assume forces and masses are measured to be the same in any reference frame for speeds small compared with the speed of light. (a) Show that Newton’s second law is also valid for an observer moving at a constant speed, small compared with the speed of light, relative to the laboratory frame. (b) Show that Newton’s second law is not valid in a reference frame moving past the laboratory frame with a constant acceleration.

(a)

Expert Solution
Check Mark
To determine

To show: The Newton’s second law is valid for an observer moving at a constant speed of light, relative to the laboratory frame.

Answer to Problem 1P

The Newton’s second law is valid for an observer moving at a constant speed of light, relative to the laboratory frame.

Explanation of Solution

Assume v is the constant speed of the frame.

The Galilean coordinate transformation is,

x=xvt

To find the velocity, take a time derivative ddt , dt=dt and v is constant.

u=dxdt

Substitute xvt for x in above equation.

u=d(xvt)dt=dxdtvd(t)dt=dxdtvd(t)dt=uv

To find the acceleration, take another time derivative ddt and dt=dt .

a=dudt

Substitute uv for u in above equation,

a=ddt(uv)a=dudtdvdta=dudta=dudt

It is shown from the above equation that the accelerations are identical.

The Newton’s second law is the same.

F=maF=ma

Thus, the Newton’s second law is valid for an observer moving at a constant speed of light, relative to the laboratory frame.

Conclusion:

Therefore, the Newton’s second law is valid for an observer moving at a constant speed of light, relative to the laboratory frame.

(b)

Expert Solution
Check Mark
To determine

The Newton’s second law is not valid in a reference frame moving past the laboratory frame with a constant acceleration.

Answer to Problem 1P

The Newton’s second law is not valid in a reference frame moving past the laboratory frame with a constant acceleration.

Explanation of Solution

Assume a0 is the constant acceleration of the frame and at time t=t=0 the two frames are together, then at some arbitrary time t later, the distance between the two frames is 12a0t2 .

The Galilean coordinate transformation is,

x=x12a0t2

To find the velocity, take a time derivative ddt , dt=dt and v is constant.

u=dxdt

Substitute x12a0t2 for x in above equation,

u=ddt(x12a0t2)=dxdta0tdtdt=ua0t

To find the acceleration, take another time derivative ddt and dt=dt .

a=dudt

Substitute ua0t for u in above equation,

a=ddt(ua0t)=dudta0dtdt=aa0

It is shown from the above equation that the accelerations are not identical.

The Newton’s second does not have same value in two different frames,

F=ma

Substitute aa0 for a in above equation,

F=m(aa0)=mama=Fma

Conclusion:

Therefore, Newton’s second law is not valid in a reference frame moving past the laboratory frame with a constant acceleration.

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Chapter 38 Solutions

Physics for Scientists and Engineers with Modern Physics

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Length contraction: the real explanation; Author: Fermilab;https://www.youtube.com/watch?v=-Poz_95_0RA;License: Standard YouTube License, CC-BY