# Relativity - velocity of object travelling perpendicular to the observer

• struggles
In summary: The answer is that there is no 'it' that goes faster than c, if 'it' refers to some definite particle or wave. A change of position is not a 'it'.In summary, the question about the relative velocity of two light beams colliding head on cannot be interpreted in the usual way due to the fact that light beams cannot have inertial reference frames. However, in the inertial reference frame of an observer standing near the point where the beams will meet, the relative change in position of the two wavefronts will be 2c. For the second part of the question, the use of time dilation can help determine the apparent velocity of a particle moving north at 0.85c relative to an observer on
struggles

## Homework Statement

1. Two light beams collide head on. Calculate their relative velocity.
2. (c) A particle moves north at speed 0.85c relative to an observer standing on the Earth. What is the velocity of this particle as observed by a fast ship traveling east on the Earth at speed 0.9c? Give the direction of travel with respect to a compass direction

## The Attempt at a Solution

1) A little confused by this question. I thought that speed of light is a constant no matter what reference frame it is viewed from so their relative velocity would be just c?

2) So speed between inertial frames u = 0.9c. Gamma γ = 1/√1-u2/c2 = 1/√0.19
However I'm unfamiliar with working with the velocity of the object perpendicular to the relative velocity between the frames.
I've found Vy' = Vy/γ(1 - uVx/c2 where the dash frame is the frame of the ship. For this question would i take Vx = 0 and Vy = 0.85c and just end up with Vy' = Vy/γ? There would then be only 'north' velocity and none in the east direction?

struggles said:
Two light beams collide head on. Calculate their relative velocity.
Your confusion is well justified. This is a weird question. Normally when we ask about relative velocity, we ask 'what is the velocity of A relative to B', which means 'in the inertial, momentarily comoving reference frame of B, what is the velocity of A'.

But light beams cannot have inertial reference frames (or any other sensible reference frames), so the question cannot be interpreted in the usual way.

We could interpret it as meaning this:
In the inertial reference frame of an observer standing near the point where the two light beams will meet, with the beams traveling along the x axis, let ##x_a(t)## and ##x_b(t)## be the x coordinates of the two wavefronts at time ##t##. Then what is ##\frac d{dt}(x_a-x_b)##?

The answer to this will be ##2c##. That doesn't break any laws of physics because the ##2c## is not the velocity of any particle, massless or otherwise, or of any wave. It is a purely abstract quantity and hence not bound by relativistic speed limits.

Buzz Bloom
andrewkirk said:
Your confusion is well justified. This is a weird question. Normally when we ask about relative velocity, we ask 'what is the velocity of A relative to B', which means 'in the inertial, momentarily comoving reference frame of B, what is the velocity of A'.

But light beams cannot have inertial reference frames (or any other sensible reference frames), so the question cannot be interpreted in the usual way.

We could interpret it as meaning this:
In the inertial reference frame of an observer standing near the point where the two light beams will meet, with the beams traveling along the x axis, let ##x_a(t)## and ##x_b(t)## be the x coordinates of the two wavefronts at time ##t##. Then what is ##\frac d{dt}(x_a-x_b)##?

The answer to this will be ##2c##. That doesn't break any laws of physics because the ##2c## is not the velocity of any particle, massless or otherwise, or of any wave. It is a purely abstract quantity and hence not bound by relativistic speed limits.

I think that kind of makes sense. So you can view the relative change in velocity as being greater than c but it doesn't actually go faster than c? Any chance of a hint of how to do the second part of my question?

struggles said:
Two light beams collide head on. Calculate their relative velocity.
I would guess that the idea here is to apply the usual relativistic sum of velocities to show that the answer is still c.

For the second part, you could use time dilation to figure out the apparent velocity of the particle in the Northerly direction. Then recombine with the ship's velocity (I think Pythagoras applies here) to get the relative velocity. The result looks sensible anyway.

struggles said:
you can view the relative change in velocity as being greater than c
It's a relative change in position (of the two wavefronts), not a relative change of velocity.
but it doesn't actually go faster than c?
One needs to choose one's words carefully in relativity. What is the 'it' referring to?

## 1. What is the theory of relativity?

The theory of relativity, developed by Albert Einstein, is a fundamental principle in physics that explains the relationship between space and time. It states that the laws of physics are the same for all observers, regardless of their relative motion.

## 2. How does the theory of relativity affect the velocity of an object travelling perpendicular to an observer?

According to the theory of relativity, the velocity of an object travelling perpendicular to an observer will appear to be different for different observers. This is because the speed of light is constant and the observers' relative motion will affect their perception of time and space.

## 3. Is the velocity of an object travelling perpendicular to an observer affected by the observer's position?

Yes, the observer's position will affect their perception of the velocity of an object travelling perpendicular to them. This is due to the phenomenon of time dilation, which states that time moves slower for objects in motion compared to those at rest.

## 4. How does the velocity of an object travelling perpendicular to an observer affect its measurement of distance?

The velocity of an object travelling perpendicular to an observer can affect its measurement of distance due to length contraction. This means that the length of an object in motion appears shorter to an observer than to an observer at rest.

## 5. Can an object travel at the speed of light perpendicular to an observer?

No, according to the theory of relativity, the speed of light is the maximum speed at which any object can travel. Therefore, an object cannot travel at the speed of light perpendicular to an observer, as it would violate the laws of physics.

• Introductory Physics Homework Help
Replies
5
Views
2K
• Introductory Physics Homework Help
Replies
20
Views
1K
• Introductory Physics Homework Help
Replies
7
Views
1K
• General Discussion
Replies
5
Views
1K
• Introductory Physics Homework Help
Replies
1
Views
1K
• Introductory Physics Homework Help
Replies
13
Views
2K
• Introductory Physics Homework Help
Replies
8
Views
1K
• Introductory Physics Homework Help
Replies
9
Views
1K
• Introductory Physics Homework Help
Replies
4
Views
1K
• Introductory Physics Homework Help
Replies
12
Views
3K