How Does Time Dilation Affect Flight Duration in Special Relativity?

[deenuh20]

Homework Statement

A plane flies with a constant velocity of 208 m/s. The clocks on the plane show that it takes exactly 2.00 hours to travel a certain distance. Calculate how much longer or shorter than 2.00 h this flight will last, according to clocks on the ground. ________s

Homework Equations

delta t = delta t(not) / (1-v^2/c^2)

The Attempt at a Solution

Because the velocity of the plane is very low, I used the binomial expansion: delta t = delta t(not)*[1 + 1/2*[208m/s / 3x10^8 m/s]^2]. When I worked it out and solved for delta t, I got .999, which I then added to the 7200s ( 2 hours). I know that the time according to the ground clocks will be longer, but I think there's a math error somewhere which is preventing me from understanding and getting the answer. Any help is appreciated. Thank you!

 

[Dick]

What are delta(t) and delta(t0) (nought not not) supposed to mean? Get rid of the deltas in that formula and realize delta(t)=t-t0. And you won't get .999 unless your calculator only has three digits of precision.

 

[m2003]

I would like to point out that this problem involves the concept of time dilation in special relativity. According to the theory of relativity, time is not absolute and can be affected by the relative velocity between two observers. In this case, the observer on the ground and the observer on the plane will have different measurements of time due to their relative velocity.

To solve this problem, we can use the formula for time dilation, which is given by delta t = delta t(not) / (1-v^2/c^2). Here, delta t is the time measured by the observer on the ground, delta t(not) is the time measured by the observer on the plane, v is the velocity of the plane, and c is the speed of light.

Substituting the given values, we get delta t = 2 hours / (1 - (208 m/s)^2 / (3x10^8 m/s)^2). Solving this, we get delta t = 2.000000000001 hours. This means that according to the ground clocks, the flight will last 0.000000000001 hours longer than 2 hours. This may seem like a very small difference, but it is a significant effect of time dilation at higher velocities.

In conclusion, the concept of time dilation is an important aspect of special relativity and should be considered when dealing with high velocities. The slight difference in time measured by the ground clocks in this problem highlights the importance of understanding this concept in our understanding of the universe.