Chapter 15, Problem 38A

You are standing facing forward on the floor of your starship, which is moving at 0.80c relative to Earth. Before you left Earth, you measured your feet to be 25 cm long.

a. People on Earth will now measure your feet to be how long?

b. Do you need to be concerned now that the shoes that you packed for the trip will be too big?

To determine

The length of the feet on the floor of starship.

Answer to Problem 38A

The length of the feet on the floor of starship is $15\text{ cm}$ .

Explanation of Solution

Given:

The length of feet is $L_0=25\text{ cm}$ .

The speed is $v=0.8c$ .

Formula used:

The expression for relativistic length contraction is,

  $L=L_0\sqrt{1-\frac{v^2}{c^2}}$

Here, $L_0$ is the measured length of the object at rest, $v$ is the speed of the object relative to an observer, and $c$ is the speed of the light.

Calculation:

The length of the feet on the floor of starship is,

  $\begin{array}{l}L=L_0\sqrt{1-\frac{v^2}{c^2}}\\ L=\left(25\text{ m}\right)\sqrt{1-\frac{{\left(0.8c\right)}^2}{c^2}}\\ L=15\text{ cm}\end{array}$

Conclusion:

Thus, the length of the feet on the floor of starship is $15\text{ cm}$ .

To determine

To explain: Whether a person should concern that the shoes packed for the trip will be too big.

Answer to Problem 38A

Yes, the shoes that you packed for the trip will be too big.

Explanation of Solution

Introduction:

Consider an object moving at a very high speed relative to an observer. Then, the length of the object seems to be contracted in the motion’s direction. The expression for relativistic length contraction is,

  $L=L_0\sqrt{1-\frac{v^2}{c^2}}$

Here, $L_0$ is the measured length of the object at rest, $v$ is the speed of the object relative to an observer, and $c$ is the speed of the light.

Calculation:

Consider a person standing inside the moving spaceship.

For the person inside the spaceship, there is no relative motion between the things which he can observe in his own frame of reference.

On the other hand, the spaceship is moving with a velocity relative to the observer on Earth.

There is a relative speed between the person on the spaceship and the frame of reference by an observer

So the feet of the person inside the spaceship seems to be contracted relative to an observer.

Hence, the shoes packed by the person for the trip will be too big.

Conclusion:

Thus, the shoes that are packed for the trip will be too big.