a)What speed must one frame of reference be moving relative to another for the two to disagree about the length by 1%
b)What speed must one frame of reference be moving relative to another for the two frames to disagree about time intervals by 1%?
Both answer should be fraction of speed of light.
a) To determine the speed at which two frames of reference disagree about length by 1%, we can use the concept of relativistic length contraction. According to special relativity, the length of an object moving relative to an observer will appear shorter than its rest length.
The formula for length contraction is given by:
L' = L * √(1 - v^2/c^2),
where:
L' is the observed length,
L is the rest length of the object,
v is the relative velocity between the frames of reference, and
c is the speed of light.
We need to find the velocity that corresponds to a 1% discrepancy, which means L' = 0.99L.
0.99L = L * √(1 - v^2/c^2),
Simplifying the equation:
0.99 = √(1 - v^2/c^2),
Squaring both sides:
0.9801 = 1 - v^2/c^2,
Rearranging the equation:
v^2/c^2 = 1 - 0.9801,
v^2/c^2 = 0.0199,
v^2 = 0.0199c^2,
Taking the square root:
v = 0.141c,
where c is the speed of light.
Therefore, the two frames of reference must have a relative velocity of approximately 0.141 times the speed of light for their length measurements to differ by 1%.
b) To determine the speed at which two frames of reference disagree about time intervals by 1%, we can use the concept of time dilation. According to special relativity, time appears to run slower for a moving observer compared to an observer at rest.
The formula for time dilation is given by:
t' = t * √(1 - v^2/c^2),
where:
t' is the observed time interval,
t is the time interval in the rest frame,
v is the relative velocity between the frames of reference, and
c is the speed of light.
We need to find the velocity that corresponds to a 1% discrepancy, which means t' = 0.99t.
0.99t = t * √(1 - v^2/c^2),
Simplifying the equation:
0.99 = √(1 - v^2/c^2),
Squaring both sides:
0.9801 = 1 - v^2/c^2,
Rearranging the equation:
v^2/c^2 = 1 - 0.9801,
v^2/c^2 = 0.0199,
v^2 = 0.0199c^2,
Taking the square root:
v = 0.141c,
where c is the speed of light.
Therefore, the two frames of reference must have a relative velocity of approximately 0.141 times the speed of light for their time interval measurements to differ by 1%.