Chapter 15, Problem 39A

Pinocchio is concerned that Gepetto will see his long hose and realize that he has been lying. So Pinocchio decides to run past Gepetto fast enough that his 10-inch long nose will be seen by Gepetto to be only 2 inches long.

a. How fast must Pinocchio run?

b. When running past Gepetto, will Pinocchio see Gepetto’s nose shortened?

To determine

The speed of person P while running.

Answer to Problem 39A

The running speed of person P is $0.98c$ .

Explanation of Solution

Given:

The length of nose is $L_0=10\text{ in}\text{.}$

The relativistic length of nose is $L=2\text{ in}\text{.}$

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 speed of personcan be calculated as,

  $\begin{array}{l}L=L_0\sqrt{1-\frac{v^2}{c^2}}\\ 2\text{ in}\text{.}=\left(\text{10 in}\right)\sqrt{1-\frac{v^2}{c^2}}\\ 1-\frac{v^2}{c^2}=0.04\\ v^2=0.96c^2\\ v=0.98c\end{array}$

Conclusion:

Thus, the running speed of person P is $0.98c$ .

To determine

To identify:Whether person Pwill see person G nose shortened.

Answer to Problem 39A

Yes, the nose of person G will appear shortened.

Explanation of Solution

Introduction:

If an object is moving at a high speed with respect to an observer, then there is a contraction in the length of the moving object in the direction of motion. The expression for relativistic length contraction is

  $L=L_0\sqrt{1-\frac{v^2}{c^2}}$

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

Calculation:

The contraction of length of a moving object depends on the time dilation.There will be no relative speed between the people on the spaceship and the things, if they observe from their own reference frame. But there will a relative speed between the person on the spaceship and an observer for different frame of reference.

From the frame of reference of person P, the nose of person G appears to be contracted. So, the nose of person G will be seen shortened to person P.

Conclusion:

Thus, the nose of person G will appear shortened to person P.