The distance to Planet X from Earth is 1.00 light year. Part A How long does it

Question

The distance to Planet X from Earth is 1.00 light year.

Part A
How long does it take a spaceship to reach X, according to the pilot of the spaceship, if the speed of the ship is 0.500c relative to X?

Delta T_1=.....y

Part B
How long does it take the ship to make the trip according to an astronaut already stationed on Planet X?

Delta T_2=.....y

Part C
Determine the distance between Earth and Planet X according to the pilot.

L1=...1y

part D
Determine the distance between Earth and Planet X according to the X-based astronaut.

L2=.....1y

Explanation / Answer

For these questions we will have to use the formula for relativistic time dilation which states the following T = GAMMA T ' Where T' is the proper time and T is the time measured by an observer in a moving reference frame and GAMMA is the velocity dependent factor = 1/sqrt(1-v^2/c^2) First, lets calculate GAMMA since it will be the same for all parts of the problem, GAMMA = 1/sqrt(1-v^2/c^2) = 1/sqrt(1-(0.5c)^2 / c^2) = 1/sqrt(0.75) = 1.155 Next we need to identify who is measuring the proper time (T'), the person measuring the proper time is the person who is moving with the moving reference frame (keep in mind that which frame is moving depends on your perspective, but in this case it is easier to think of the person on planet X as stationary and the astronaut as in the moving reference frame) Therefore, T' is the time measured by the astronaut. It is actually easier to solve part B first, from the perspective of the person on planet X, he simply sees an object moving with a constant speed over a certain distance, so the time he measures is given by the simple formula, t = d/v t= 1 light year / 0.5 c t = 1 year*c / 0.5 c t = 2 years Note that we didn't have to use any dilation formulas for this because the observer was in a stationary reference frame and was simply measuring the speed of a moving object relative to a rest frame. Now we can calculate the time that the astronaut thought passed using the time dilation formula I gave above, T = GAMMA T ' We are looking for the time for the astronaut in the moving reference frame, so we want T ', so solving for this gives us T' = T/GAMMA Plug in the value we got for T and gamma to get T ' = 2 years / 1.155 = 1.732 years As we would expect, the person moving relative to the rest frame experienced less time passage than the person at rest. Now, as far as the distance that each person measured for the journey, this can be done either by using the speed and the time each of them measured, or by using the length dilation formula, i will show both just to show that they give consistent results. The person on planet X measured the distance to be 1 light year, this was given. Now we can calculate the distance the astronaut measured again with the formula d = v*t, the only caveat here is that you need to use the dilated time we calculated, The astronaut measures the solar system moving at 0.5 c and he measures that the elapsed time equals 1.732 years, so he measures a distance of d = 0.5 c * 1.732 years = 0.866 light years Now lets use the formula for length dilation and check that it matches, for this we have the formula L = L' / GAMMA Here is where you may get confused, before the proper reference frame was the astronaut in the ship, but here the proper frame is the person on planet X, this is because the proper frame for length is the frame which is at rest relative to the length. This is important so I will state both here explicitly *** The proper time is the time measured in the reference frame in which two events occur in the same place, i.e. the frame in which a clock would appear to be in the same spot, like on a shelf on the space ship *** The proper length is the length measured from the reference frame that is not moving relative to the length being measured So the proper length is the length measured by the person on planet X, or 1 light year and L is the length measured by the astronaut. We plug in the values for L ' a GAMMA L = L' / GAMMA L = 1 light year / 1.155 L = 0.866 light years As you can see these two methods give matching values, the only tricky part about problems like these is identifying what to plug in where, how to identify T vs T' and L vs L' , but I hope these definitions I gave you can help make it a bit clearer!

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