If a clock at rest is three times faster than a moving clock, we can determine the speed of the moving clock using the concept of time dilation from special relativity.
According to special relativity, the time measured by a moving clock appears to be dilated or stretched out compared to the time measured by a clock at rest relative to the observer. The time dilation factor (γ) is given by the equation:
γ = 1 / √(1 - v^2/c^2)
Where:
- v is the velocity of the moving clock relative to the observer.
- c is the speed of light in a vacuum (approximately 299,792,458 meters per second).
Let's assume the clock at rest measures a unit of time (let's say seconds) and the moving clock measures t seconds. According to the given information, the clock at rest is three times faster, so t = 1/3.
Plugging these values into the time dilation equation, we have:
1/3 = 1 / √(1 - v^2/c^2)
Squaring both sides of the equation, we get:
1/9 = 1 / (1 - v^2/c^2)
Rearranging the equation, we have:
1 - v^2/c^2 = 9
v^2/c^2 = 1 - 1/9
v^2/c^2 = 8/9
Taking the square root of both sides, we have:
v/c = √(8/9)
v/c ≈ 0.94281
To find the speed of the moving clock, we multiply this result by the speed of light:
v = (0.94281) * c
v ≈ (0.94281) * 299,792,458 m/s
v ≈ 282,743,338 m/s
Therefore, the speed of the moving clock is approximately 282,743,338 meters per second.