A box is cubical with sides of proper lengths L1=L2=L3=2.0m when viewed in it own rest frame. If this block moves parallel to one of its edges with a speed of 0.80c past an observer,
(a) What shape does it appear to have to this observer, and
(b) What is the length of each side as measured by the observer?
The edges parallel to the direction the box travels will have shortened by a factor
1/gamma = sqrt[1-v^2/c^2] = 0.6
To answer the questions:
(a) When the box moves parallel to one of its edges with a speed of 0.80c past an observer, it will appear to be length-contracted in the direction of motion. This means that it will appear to be flattened and have a different shape to the observer.
(b) The length of each side as measured by the observer can be calculated using the formula for length contraction. Since the edges parallel to the direction of motion have shortened by a factor of 0.6 (1/gamma), the length of each side as measured by the observer would be:
L' = L * 0.6
Substituting the value of L (2.0m) into the equation:
L' = 2.0m * 0.6
L' = 1.2m
Therefore, each side of the box will appear to be 1.2m in length to the observer.
To answer these questions, we need to use the concept of length contraction, which occurs when an object moves at relativistic speeds. Length contraction means that an object appears shorter in the direction of its motion when viewed from a different reference frame.
(a) To determine the shape the box appears to have to the observer, we need to consider that only the edges parallel to the direction of motion will be affected by length contraction. The edges perpendicular to the direction of motion will not change. Since all the sides of the box are initially equal, after length contraction, the box will have the appearance of a rectangular prism.
(b) To find the length of each side as measured by the observer, we need to multiply the proper length of each side by the length contraction factor, which is given by 1/gamma. In this case, the length contraction factor is 0.6.
So, if the proper length of each side L1, L2, L3 is initially 2.0 meters, the length of each side as measured by the observer will be:
L1' = L1 * 0.6 = 2.0 * 0.6 = 1.2 meters
L2' = L2 * 0.6 = 2.0 * 0.6 = 1.2 meters
L3' = L3 = 2.0 meters
Therefore, the length of each side of the box, as measured by the observer, will be 1.2 meters for two sides parallel to the direction of motion, and 2.0 meters for the side perpendicular to the direction of motion.