(a)Explain the term time dilation.
(b) Show that an interval of time observed in a moving frame of reference will be less than the same interval observed in a stationary frame of reference.
(a) Time dilation is a concept in physics that occurs due to the theory of relativity. It states that time can appear to run slower for objects that are moving relative to an observer's reference frame.
According to the theory of relativity, time is not absolute but is relative to an observer's motion and gravitational field. Time dilation occurs because the speed of light is constant for all observers, regardless of their relative motion. As an object approaches the speed of light, time for that object appears to slow down from the perspective of an observer at rest.
In simpler terms, if two objects are moving relative to each other, time for the moving object appears to pass more slowly compared to the stationary object. This means that clocks on a moving object will run slower, and processes such as aging or any time-based measurements will be affected.
(b) To understand how an interval of time observed in a moving frame of reference is less than the same interval observed in a stationary frame of reference, we can use the concept of time dilation.
Let's consider two observers, one in a stationary frame of reference (observer A) and the other in a moving frame of reference (observer B). Observer B is moving relative to observer A at a constant velocity.
According to the theory of relativity, as the relative velocity between the two observers increases, time dilation occurs. The moving observer, B, will perceive time to be running slower compared to the stationary observer, A.
To demonstrate this, let's assume observer A measures an interval of time ΔtA between two events. Observer B, who is moving relative to A, will measure an interval of time ΔtB between the same two events.
By applying the principle of time dilation, the interval of time observed by B will be less than the interval observed by A. Mathematically, we can express this relationship as:
ΔtB = ΔtA * √(1 - (v^2 / c^2))
Where ΔtB is the interval of time observed by observer B, ΔtA is the interval of time observed by observer A, v is the relative velocity between the two observers, and c is the speed of light.
This equation shows that as the relative velocity v increases, the term (v^2 / c^2) becomes larger, causing the square root term to decrease. As a result, ΔtB becomes smaller than ΔtA, indicating that observer B measures a smaller interval of time between the same events.
This equation demonstrates how time dilation causes the measured time interval in a moving frame of reference to be less than the same interval observed in a stationary frame of reference.