Physics
Physics
5th Edition
ISBN: 9781260486919
Author: GIAMBATTISTA
Publisher: MCG
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Chapter 26, Problem 15P

(a)

To determine

The length measured in the rest frame of the particle, if the length of the field between goal lines in the earth frame is 91.5m.

(a)

Expert Solution
Check Mark

Answer to Problem 15P

The length measured in the rest frame of the particle is 79m .

Explanation of Solution

Since the length of the field between the goal lines in the rest frame of earth is 91.5m, proper length is 91.5m.

Write the expression for the length of field between the goal lines as measured in the rest frame of particle.

  L=L0γ                                                                                                                      (I)

Here, L0 is the proper length, L is the length of field between the goal lines as measured in the rest frame of particle and γ is the Lorentz factor.

Write the expression for the Lorentz factor.

  γ=11v2c2

Here, v is the speed of cosmic ray particle and c is the speed of light in vacuum.

Substitute 11v2c2 for γ in equation (I) to get L.

  L=L01v2c2                                                                                                         (II)

Conclusion:

Substitute 91.5m for L0 and 0.50c for v in equation (II) to get L.

  L=(91.5m)1(0.50c)2c2=91.5m(0.86)=79.24m79m

Therefore, the length measured in the rest frame of the particle is 79m .

(b)

To determine

The time the particle takes to go from one goal line to the other according to Earth observers.

(b)

Expert Solution
Check Mark

Answer to Problem 15P

The time the particle takes to go from one goal line to the other according to Earth observers is 610ns .

Explanation of Solution

The speed of the particle is constant with respect to earth. Thus, the time can be found using the equation Δx=vΔt, where Δx is the distance between the goal lines in frame of earth and Δt is the time taken by the cosmic ray to move from one goal line to another.

Write the expression for the time taken by the particle to move from one goal line to another relative to earth observer.

  Δt=Δxv                                                                                                                 (III)

Substitute 91.5m for Δx and 0.50c for v in equation (III) to get Δt.

  Δt=91.5m0.50c

Conclusion:

Substitute 3.00×108m/s for c in above equation to get Δt.

  Δt=91.5m0.50(3.00×108m/s)=6.1×107s×109ns1s=610ns

Therefore, the time the particle takes to go from one goal line to the other with respect to Earth observers is 610ns .

(c)

To determine

The time the particle takes to go from one goal line to the other in the rest frame of the particle.

(c)

Expert Solution
Check Mark

Answer to Problem 15P

The time the particle takes to go from one goal line to the other in the rest frame of the particle is 530ns .

Explanation of Solution

In rest frame of the particle, the distance travelled is 79m and speed is 0.50c.

Write the expression for the time required to travel between goal lines in the rest frame of the particle.

  Δt=Δxv                                                                                                                  (IV)

Here, Δx is the distance between the goal lines in rest frame the particle and Δt is the time taken by the cosmic ray to move from one goal line to another in rest frame of particle.

Conclusion:

Substitute 79m for Δx and 0.50c for v in equation (IV) to get Δt.

  Δt=79m0.50c

Substitute 3.00×108m/s for c in above equation to get Δt.

  Δt=79m0.50(3.00×108m/s)=5.3×107s×109ns1s=530ns

Therefore, the time the particle takes to go from one goal line to the other with respect to rest frame of the particle is 530ns .

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Chapter 26 Solutions

Physics

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