Ships passing each other at near light speed

[Skaperen]

TL;DR

Two ships at more than half light speed pass each other in opposite directions. It might seem their relative speed exceeds the speed of light. What is their relative speed if they go at 3/4 the speed of light?

Two spaceships line up to go past each other at a near distance (just far enough to not hit each other). A third ship is stationary near where the other two will go past each other to view the event. The two ships that will go past each other will be traveling at 3/4 (75%) of the speed of light. The speed of light is 299,792,458 meters per second, so the two ships will be traveling at 224,844.343.5 meters per second, each. How fast will they be going relative to each other? What speed will the third ship see?

 

[weirdoguy]

How fast will they be going relative to each other? What speed will the third ship see?

Google "relativistic velocity addition".

 

[FactChecker]

When you apply relativistic velocity addition equations, remember to be very specific about whose inertial reference frame each speed is measured in. Many problems would have something like a train traveling at 3/4 c and a ball thrown in the train at 3/4 c measured in the train's time and distance.

I interpret your question this way. A third observer is measuring two ships approaching each other, each at 3/4 c in the third ship's reference frame. The third ship would simply calculate that those two other ships have a closure speed of 1.5 c. But that is not what either moving ship would measure as the closure speed with the other moving ship. Their measurement would be under c.

 

[Ibix]

You need two separate concepts, relative velocity and separation rate. Relative velocity is the speed one object measures another to have using instruments at rest with respect to itself. Separation rate is the rate of change of the distance between two objects measured by a third object that may see them both in motion. In Newtonian physics these two things are the same for 1d motion (like in your setup), but not in relativity.

The separation rate is just the vector sum of the velocities, so the distance the third ship measures between the two (assuming the OP meant 0.75c relative to the third ship - it wasn't specified) will indeed change at 1.5c. That's fine - nothing is moving faster than c.

The first two ships will not measure each other to be doing 1.5c. All three ships see the others' rulers and clocks in relative motion, so length contracted, time dilated, and (critically!) not synchronised correctly. So they should not be surprised that the relative velocity measured by one of the others is not the separation rate that they measure. (The unique exception to this is the separation rate between an object a ship measures as stationary and something else; in this case the separation rate and velocity are equal because the stationary object's rulers and clocks are also stationary.)

 

[cianfa72]

You need two separate concepts, relative velocity and separation rate. Relative velocity is the speed one object measures another to have using instruments at rest with respect to itself. Separation rate is the rate of change of the distance between two objects measured by a third object that may see them both in motion.

The separation rate is just the vector sum of the velocities, so the distance the third ship measures between the two

Even in the context of relativity, relative vector velocities entering the definition of separation rate as vector sum, are not 4-velocities, right?

 

[Ibix]

Even in the context of relativity, relative vector velocities entering the definition of separation rate as vector sum, are not 4-velocities, right?

Of course