At what speed must one frame of reference be moving relative to another for the two frames to disagree about lengths by 1%? Express your result as a fraction of the speed of light.
At what speed must one frame of reference be moving relative to another for the two frames to disagree about time intervals by 1%? Express your result as a fraction of the speed of light.
To answer these questions, we will use the concepts of length contraction and time dilation, which are part of Einstein's theory of relativity.
1. Disagreement about lengths by 1%:
According to length contraction, when an object is moving relative to an observer, its length appears shorter in the direction of motion. The magnitude of this contraction is given by the Lorentz factor, which can be expressed as:
Lorentz factor = sqrt(1 - (v^2/c^2)),
where v is the relative velocity between the two frames of reference and c is the speed of light.
To find the speed at which the two frames disagree about lengths by 1%, we want the Lorentz factor to be equal to 0.99 (1% less than 1). So we can write:
sqrt(1 - (v^2/c^2)) = 0.99.
Solving this equation for v, we find:
v/c = sqrt(1 - 0.99^2),
v/c = sqrt(1 - 0.9801),
v/c = sqrt(0.0199),
v/c = 0.141,
v = 0.141c.
Therefore, the frames must be moving relative to each other at a speed of approximately 0.141 times the speed of light in order to disagree about lengths by 1%.
2. Disagreement about time intervals by 1%:
According to time dilation, when an object is moving relative to an observer, the time experienced by the moving object appears dilated compared to that experienced by the observer. The magnitude of this dilation is also given by the Lorentz factor.
Using the same Lorentz factor as before, we can write:
sqrt(1 - (v^2/c^2)) = 0.99.
Solving for v, we find:
v/c = sqrt(1 - 0.99^2),
v/c = sqrt(1 - 0.9801),
v/c = sqrt(0.0199),
v/c = 0.141,
v = 0.141c.
Again, we get the same result. This means that the frames must be moving relative to each other at a speed of approximately 0.141 times the speed of light in order to disagree about time intervals by 1%.
In both cases, the speed at which the disagreement occurs is approximately 0.141 times the speed of light.