Calculating Time for Dart to Reach Front of Moving Rocket

[w3390]

Homework Statement

A rocket with that has a proper length of 1100 m moves away from a space station in the + x direction at 0.60c relative to an observer on the station. An astronaut stands at the rear of the rocket and fires a dart toward the front of the rocket at 0.85c relative to the rocket. How long does it take the dart to reach the front of the rocket as measured in the reference frame of the space station?

Homework Equations

u=[tex]\frac{u'+v}{1+\frac{v*u'}{c^2}}[/tex]

The Attempt at a Solution

I worked through the algebra of the above equation by assigning u to the velocity of the dart relative to the space station, u' to the velocity of the dart relative to the rocket, and v to the velocity of the rocket relative to the space station. The result was that the dart is traveling at .96c relative to the space station. Since I have found the speed of the dart in the frame of the space station, do I then have to find how much the rocket contracts and then determine the time it takes to travel or I am wrong in this line of logic? If this logic is correct, the answer I got was 3.8 microseconds by dividing the proper length of the rocket by the velocity of the dart.

Thanks for any help.

 

[kuruman]

An easier way to look at the problem that does not involve addition of velocities is this. Assume that you are the astronaut firing the dart. You see it cover a distance of 1100 m traveling at speed 0.85 c. You can certainly calculate how long it takes the dart to reach the other end as measured by you. Call this time t'. What is t' as measured by an observer on the station?

 

[w3390]

Okay, I understand what you are saying and I'm thinking that the equation to use is:

t'=[tex]\frac{t-\frac{v*x}{c^2}}{\sqrt{1-\frac{v^2}{c^2}}}[/tex]

However, I am confused about what to input for the value of x. Could I set x equal to zero and solve for t?

 

[kuruman]

Use the time dilation equation,

[tex]
\Delta t=\frac{\Delta t'}{\sqrt{1-\frac{v^2}{c^2}}}
[/tex]

which says that the interval measured by the station clock is longer than the interval measured by the spaceship clock, i.e. the station clock runs faster. I put in the capital deltas to indicate that we are talking about time intervals and not the coordinate of time which is what your equation is about.

 

[w3390]

Okay thanks a lot. So just to see if I have it straight, if I know the space and time components of one event and have to figure out the space and time components of that event in a different frame, I should use the Lorentz transformations like I tried to do earlier. And if I do not know the location of an event but just want to find how long one event took in a different frame, I can use time dilation.

 

[w3390]

I have another question related to the same problem statement I gave at the beginning of the thread. This time I have to figure out how long it takes the dart to travel the length of the rocket as measured in the frame of the dart. I was certain I had it correct but I was wrong. For this problem I used the equation: t'=[tex]\frac{t-\frac{vx}{c^2}}{\sqrt{1-\frac{v^2}{c^2}}}[/tex], and assigned t= 4.3 microseconds and v=.85c. I am still confused about what I am supposed to use in place of x. This keeps coming up, I guess this is not the correct equation. However, I assigned the value of x to 1100m. I am almost certain this is where my error lies, but I cannot figure out an alternative way to do the problem. Any help on correct formulas would be greatly appreciated.

 

[kuruman]

In the frame of the dart, the length of the room moves at 0.85, but is Lorentz-contracted. To find the time, divide the contracted length by 0.85 c