How Long Does It Take for One Rocket to Pass Another at Relativistic Speeds?

[ZanyCat]

I do have a specific example/problem, but my actual question is more so conceptual (I'm sure that seeing someone confused by relativity is a first around here!).

The problem:

Two rockets are each 1000m long in their rest frame. Rocket Orion, traveling at 0.900c relative to the earth, is overtaking rocket Sirius, which is poking along at a mere 0.700c. According to the crew on Sirius, how long does Orion take to completely pass?
That is, how long is it from the instant the nose of Orion is at the tail of Sirius until the tail of Orion is at the nose of Sirius?
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Okay, so I'm thinking that I'll need to find a) the observed velocity of O in S's FOR and b) the observed length of O in S's FOR, and go from there.

My confusion is coming from the velocities. I'm taking S as my Frame of Reference, so from S's FOR, is O moving past at 0.200c? Or do I need to transform the velocity?

 

[Doc Al]

Okay, so I'm thinking that I'll need to find a) the observed velocity of O in S's FOR and b) the observed length of O in S's FOR, and go from there.

Good.

My confusion is coming from the velocities. I'm taking S as my Frame of Reference, so from S's FOR, is O moving past at 0.200c?

No. To find the velocity of O with respect to S, you'll need to use the relativistic addition of velocity formula.

 

[ZanyCat]

I'm struggling with that part. Do I need to consider the Earth as my stationary FOR, then?

 

[ZanyCat]

Never mind, got it sorted, thanks Doc!

 

[Nickie]

First, let's clarify some concepts before addressing the specific problem. Relativistic velocities refer to velocities that are significant enough to require the use of Einstein's theory of relativity to accurately describe them. This is necessary when objects are moving at speeds close to the speed of light, which is the fastest speed possible in the universe. At these speeds, classical Newtonian mechanics no longer holds true and we must take into account the effects of time dilation and length contraction.

Now, to address the specific problem, we need to use the concept of relative velocity. This is the velocity of one object as observed from the perspective of another object. In this case, the two rockets are moving relative to each other, so we need to determine their relative velocity.

To do this, we can use the formula for adding velocities in special relativity, which is given by:

v = (u + v)/(1 + uv/c^2)

Where:
v = relative velocity between the two objects
u = velocity of one object
v = velocity of the other object
c = speed of light

In this problem, we are given the velocities of the two rockets, 0.900c and 0.700c. Plugging these values into the formula, we get a relative velocity of 0.9945c. This means that from the perspective of Sirius, Orion is moving at a speed of 0.9945 times the speed of light.

Now, to find the time it takes for Orion to pass Sirius, we need to use the concept of time dilation. This is the effect where time appears to pass slower for an object that is moving at high speeds relative to another object. The formula for time dilation is given by:

t = t0/√(1 - v^2/c^2)

Where:
t = time observed by Sirius
t0 = time in the rest frame of Orion
v = relative velocity between the two objects
c = speed of light

In this case, we know that the length of Orion is 1000m in its rest frame, so t0 = 1000m/c. Plugging in the values, we get t = 1732.05m/c. This means that from the perspective of Sirius, it takes 1732.05 meters of distance for Orion to completely pass.

In conclusion, the time it takes for Orion to pass Sirius is 1732.05 meters divided by the