Question

A sailboat is manufactured so that the mast leans at an angle with respect to the deck. An observer standing on a dock sees the boat go by at speed v (Fig. 12.14). What angle does this observer say the mast makes?

Short answer

The angle observed by an observed on the dock is $\tan \overline{\theta }=\frac{\tan \theta }{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}$.

Step-by-step solution

Expression for the vertical and horizontal component of length:

Let l be the length of the mast.

From the given figure, write the x and y components of the length of the mast.

$$\begin{gathered} l_x=l\cos \theta \\ {l}_y=l\sin \theta \end{gathered}$$

As the motion of the boat is along the x-direction, the length contracted will be observed only along the x-direction while the y-component of the length remains unchanged.

Write the horizontal component of length $l_x^{\text{'}}$.

${\mathit{l}}_{\mathbf{x}}^{\mathbf{\text{'}}}\mathbf{=}\frac{\mathbf{l}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{\theta }}{\mathbf{Y}}$

Here, $y=\frac{1}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}$.

Write the vertical component of length $l_y^{\text{'}}$.

$l_y^{\text{'}}=l\sin \theta$

Determine the angle observed by an observer on the dock:

Write the expression for the angle $\overline{\theta }$observed by an observer on the dock.

$\tan \overline{\theta }=\frac{l_y^{\text{'}}}{l_x^{\text{'}}}$

Substitute the value of $l_y^{\text{'}}$and $l_x^{\text{'}}$in the above expression.

$$\begin{gathered} \tan \overline{\theta }=\frac{l\sin \theta }{\left({\displaystyle \frac{l\cos \theta }{\gamma }}\right)} \\ \tan \overline{\theta }=\frac{l\sin \theta }{\left(\frac{l\cos \theta }{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}\right)} \\ \tan \overline{\theta }=\frac{\tan \theta }{\sqrt{1-\frac{v^2}{c^2}}} \end{gathered}$$

Therefore, the angle observed by an observed on the dock is $\overline{\theta }=\frac{\tan \theta }{\sqrt{1-\frac{v^2}{c^2}}}$.