Chapter 1.5, Problem 1aT

The second diagram at right shows the positions of spaceships A, B, and the shuttle at time $t_i$ in the reference frame of spaceship B.

Sketch spaceship A and the shuttle at their positions at time $t_f$ as measured in the reference frame of spaceship B.

Explain how the diagram is consistent with the fact that in its own frame of reference, spaceship B is not moving.

Diagram for refrence frame of spaceship B

In the box at right, draw and label vectors for the following quantities:

${\stackrel{\to }{x}}_S^{(i)}{\stackrel{\to }{x}}_S^{(f)},\text{and}\Delta {\stackrel{\to }{x}}_S$ in frame of spaceship B

• ${\stackrel{\to }{x}}_{S,B}{}^{(i)}$ (the initial position of the shuttle in the frame of spaceship B)
• ${\stackrel{\to }{x}}_{S,B}{}^{(f)}$ (the final position of the shuttle in the frame of spaceship B)
• ${\stackrel{\to }{x}}_{S,B}$ (the displacement, or change in position, of the shuttle in the frame of spaceship B).

Is the quantity ${\stackrel{\to }{x}}_{S,B}$ associated with:

• a single instant in time or an interval of time? Explain.

• the distance between two objects or the distance traveled by a single object? Explain.

Describe how you could use $\Delta {\stackrel{\to }{x}}_{S,B}$ , to determine the velocity of the shuttle in the frame of spaceship B.

To determine

To Draw: Spaceship A and the shuttle at their positions at time $t_f$ in the reference frame of spaceship B.

The vectors of the following quantities in frame of spaceship B ${\overrightarrow{x}}_{\text{S,B}}{}^{\left(\text{i}\right)}$ , ${\overrightarrow{x}}_{\text{S,B}}{}^{\left(\text{f}\right)}$ and $\Delta {\overrightarrow{x}}_{\text{S,B}}$ . The association of quantity change in position ( $\Delta {\overrightarrow{x}}_{\text{S,B}}$ ) with time and distance.

To Explain:Whether the diagram is consistent with the fact that in its own frame of reference spaceship B is not moving.

The use of $\Delta {\overrightarrow{x}}_{\text{S,B}}$ in finding the velocity of the shuttle in the frame of spaceship B.

Answer to Problem 1aT

The diagram is consistent with the fact that in its own frame of reference spaceship B is not moving. The quantity $\Delta {\overrightarrow{x}}_{\text{S,B}}$ is associated with interval of time and it represents distance travelled by single object.

Explanation of Solution

Introduction:

Spaceship A and B are moving towards each other and spaceship A releases the shuttle at time $t_i$ . Spaceship A, B and the shuttle at time $t_i$ is given.

Spaceship A and the shuttle at their positions at time $t_f$ as measured in the reference frame of spaceship B is represented in the diagram given below.

  

The diagram is consistent with the fact that in its own frame of reference spaceship B is not moving as the body appears to be at rest with its own frame of reference.

Diagram having following quantities ${\overrightarrow{x}}_{\text{S,B}}{}^{\left(\text{i}\right)}$ , ${\overrightarrow{x}}_{\text{S,B}}{}^{\left(\text{f}\right)}$ and $\Delta {\overrightarrow{x}}_{\text{S,B}}$ in frame of spaceship B is drawn below.

  

Here, ${\overrightarrow{x}}_{\text{S,B}}{}^{\left(\text{i}\right)}$ is the initial position of the shuttle in the frame of spaceship B, ${\overrightarrow{x}}_{\text{S,B}}{}^{\left(\text{f}\right)}$ is the final position of the shuttle in the frame of spaceship B and $\Delta {\overrightarrow{x}}_{\text{S,B}}$ is the displacement or change in position of the shuttle in the frame of spaceship B.

The quantity $\Delta {\overrightarrow{x}}_{\text{S,B}}$ is associated with interval of time and it represents distance travelled by single object. It can be used to find velocity of shuttle in the frame of spaceship B using the expression given below.

  $\overrightarrow{v}=\frac{\Delta {\overrightarrow{x}}_{\text{S,B}}}{t_f-t_i}$

Here, $t_f$ is the final position of the shuttle and $t_i$ is the initial position of the shuttle.

Conclusion:

Thus, the diagram is consistent with the fact that in its own frame of reference spaceship B is not moving. The quantity $\Delta {\overrightarrow{x}}_{\text{S,B}}$ is associated with interval of time and it represents distance travelled by single object.