Question

A suitcase kept on a shop's rack is measured 50 cm × 25 cm × 10 cm by the shop's owner. A traveller takes this suitcase in train moving with velocity 0.6c. If the suitcase is placed with its length along the train's velocity, find the dimensions measured by (a) the traveller and (b) a ground observer.

Answer 1

Given:
Length of suitcase, l = 50 cm
Breadth of suitcase, b = 25 cm
Height of suitcase, h = 10 cm
Velocity of train, v = 0.6c

(a) The observer in the train notices the same values of l, b and h because the suitcase is in rest w.r.t. the traveller.

(b) Since the suitcase is moving with a speed of 0.6c w.r.t. the ground observer, the component of length parallel to the velocity undergoes contraction, but the perpendicular components (breadth and height) remain the same.

So, the length that is parallel to the velocity of the train changes, while the breadth and height remain the same.

$$\begin{gathered} l\text{'}=l\sqrt{1-\frac{v^2}{c^2}} \\ =50\sqrt{1-\frac{{\left(0.6c\right)}^2}{c^2}} \\ =50\sqrt{1-0.36} \\ =50\sqrt{0.64} \\ =50\times 0.8=40\mathrm{cm} \end{gathered}$$

Thus, the dimensions measured by the ground observer are 40 cm × 25 cm × 10 cm.