How Fast Is the Spaceship Traveling Relative to Earth in Terms of c?

[map7s]

Homework Statement

As measured in Earth's frame of reference, two planets are 491,000 km apart. A spaceship flies from one planet to the other with a constant velocity, and the clocks on the ship show that the trip lasts only 1.10 s. How fast is the ship traveling? (Enter your answer to 4 significant figures and in terms of c.)

Homework Equations

I tried using the equation delta t=d/v and converting the km to m, but that seemed way to easy...and I got the wrong answer. Then I tried delta t=delta t o / sqrt(1-v^2/c^2), but I wasn't sure what to plug in for the variables. Assuming that I am now on the right path, would I put in 1.1 for delta t? Would I even need anything for delta t o ?

 

[Nate810]

What about for d, would I put in 491,000?The Attempt at a SolutionI tried delta t=delta t o / sqrt(1-v^2/c^2), with delta t o =1.1, d=491,000 and v unknown. Solving for v, I got 4.7*10^6 m/s. Converting to c, I got 0.015 c.

 

[m2003]

As a scientist, it is important to approach problems with a systematic and logical approach. Let's break down the given information and use the appropriate equations to solve this problem.

Firstly, we are given the distance between the two planets, 491,000 km, and the time it takes for the spaceship to travel this distance, 1.10 s. We can use the equation d = vt, where d is distance, v is velocity, and t is time, to solve for the velocity of the spaceship.

Next, we know that the clocks on the spaceship show a different time than the clocks on Earth. This is due to the effects of time dilation, a concept explained by Einstein's theory of relativity. The equation for time dilation is t = t0 / √(1-v^2/c^2), where t0 is the time measured in the stationary frame of reference (in this case, Earth) and t is the time measured in the moving frame of reference (the spaceship).

To solve for the velocity, we can set these two equations equal to each other and solve for v.

d = vt

t = t0 / √(1-v^2/c^2)

Substituting the given values, we get:

491,000 km = v(1.10 s)

1.10 s = t0 / √(1-v^2/c^2)

Solving for v, we get:

v = (491,000 km/1.10 s) / (1 / √(1-v^2/c^2))

v = 446,363,636.36 km/s

Converting this to meters per second and dividing by the speed of light, c, which is 299,792,458 m/s, we get:

v = 0.149 c

Therefore, the spaceship is traveling at a speed of 0.149 times the speed of light.

It is important to note that this is the speed of the spaceship relative to Earth's frame of reference. The actual speed of the spaceship may be different depending on the reference frame used. Also, it is important to check the significant figures in the final answer and round accordingly.

In conclusion, using the appropriate equations and understanding the effects of relativity, we can solve this homework problem and determine the speed of the spaceship in terms of c. I hope this