If an observer is traveling parallel to a rod at twice the speed of the rod, what is does the observer perceive the length to be?
Choose one answer.
a. The length of the rod remains the same.
b. The length of the rod will become twice its original value.
c. The length of the rod will become half of the original value.
d. The length of the rod will decrease.
To determine what the observer perceives the length of the rod to be, we can consider the concept of length contraction, which is a consequence of special relativity.
According to special relativity, when an object is moving relative to an observer, its length appears contracted in the direction of motion. This effect is more pronounced as the relative speed between the observer and the object increases.
In this case, the observer is traveling parallel to the rod at twice the speed of the rod. As a result, the observer's relative speed to the rod is relatively high.
The observer will perceive the length of the rod to be contracted. The amount of contraction can be calculated using the Lorentz transformation formula:
L' = L *√(1 - v^2/c^2)
Where:
L' is the perceived length of the rod,
L is the original length of the rod,
v is the relative velocity between the observer and the rod, and
c is the speed of light.
Since the observer's relative speed is twice that of the rod, we can substitute v = 2v into the formula:
L' = L *√(1 - (2v)^2/c^2)
= L *√(1 - 4v^2/c^2)
Since v is a positive quantity (the observer is traveling parallel to the rod), v^2 will be positive. The square of a positive quantity is always positive, so 4v^2/c^2 will be positive as well.
The expression inside the square root will therefore be less than 1, resulting in a contraction of the length. Therefore, the correct answer is:
d. The length of the rod will decrease.