Einstein velocity transformations problem

In summary: There are a few typos on the site, but nothing fatal.In summary, the conversation discusses the use of Einstein velocity transformations between inertial frames to determine whether a bird flying at a constant velocity of .5c is heading towards a birdfeeder located 1 light-minute north and .4 light-minutes east of the observer's position. The calculations lead to a contradiction, but it is resolved by taking into account the effects of space contraction.
  • #1
Sleepycoaster
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0
So I made this problem up to visualize the einstein velocity transformations between inertial frames.

Homework Statement



I throw a frisbee due north. It goes north at a constant velocity of .7c. At the same time I throw it, a bird flies in a straight line at a constant velocity of .5c at such an angle that its northward component is .3c and its eastward component is .4c, relative to the frisbee. It is going toward a birdfeeder located 1 light-minute north and .4 light-minutes east of where I stand. Is the bird really going toward the birdfeeder in both the frisbee's inertial frame and my inertial frame?

Homework Equations



vx = (vx' + β)/(1+vx'β)
vy = (vy'(√1 - β2)/(1-vx'β)

Apostrophied velocities are measured in the frisbee's frame, which moves at velocity "beta" relative to my frame.

The Attempt at a Solution



In the frisbee's frame, the birdfeeder is heading south at .7c. In one minute, it will be .7 light-minutes south of where it was before. The bird moves relative to the frisbee up .3 light-minutes and east .4 light-minutes, so it should meet the birdfeeder in one minute.

In the home frame,

The northward component of the bird is .3+.7 / 1+(.3)(.7) = .82645
The eastward component of the bird is .4(sqrt(1-.49))/1+(.3)(.7) = .23608

Since .82645/.23608 does not equal 1/.4, the bird is not heading toward the birdfeeder.

I definitely did something wrong to get this contradiction. Would anyone like to try it?
 
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  • #2
Is the bird really going toward the birdfeeder in both the frisbee's inertial frame and my inertial frame?
... to word "really" does not belong here.

You can check your setup by sketching the space-time diagrams for each observer.
 
  • #3
You forgot about space contraction which changes the angle towards the location of the bird feeder.
 
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  • #4
Dauto,

Thanks for the reply, but I'm not sure I get it. Wouldn't the Einstein velocity transformations already account for the space contraction between the points of view of me and the frisbee?
 
  • #5
Okay, I figured it out. From my point of view, the bird flew north at .82645c, but if I were to use a simple Lorentz contraction and multiply this velocity by sqrt(1-(.7)^2), you get a velocity of .5902c north, which, coupled with the .23608c component East, will get the bird to the birdfeeder.
 
  • #6
Well done - it's easier with the space-time diagrams though.
 
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  • #7
I'm afraid I don't know space-time diagrams very well. Do you mean, a graph with perpendicular axes "time" and "distance" with a second "time" axis at slope 1/.7 and a second "distance" axis at slope .7, all from the origin?
 
  • #8
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What is the Einstein velocity transformations problem?

The Einstein velocity transformations problem is a thought experiment proposed by Albert Einstein in his theory of special relativity. It involves two observers moving at different velocities relative to each other and how they would perceive and measure the same event.

Why is the Einstein velocity transformations problem important?

The Einstein velocity transformations problem is important because it challenges our understanding of space and time. It led to the development of the theory of special relativity, which has had a significant impact on modern physics and our understanding of the universe.

How are the Einstein velocity transformations solved?

The Einstein velocity transformations are solved using mathematical equations that describe the relationship between space and time for observers moving at different velocities. These equations take into account the speed of light, which is constant for all observers, and allow for the transformation of measurements between different reference frames.

What is the Lorentz factor and how does it relate to the Einstein velocity transformations?

The Lorentz factor is a mathematical term used in the equations of special relativity to describe the dilation of time and contraction of space as an object approaches the speed of light. It is a key factor in the Einstein velocity transformations, as it allows for the correct transformation of measurements between reference frames.

What are some real-world applications of the Einstein velocity transformations?

The Einstein velocity transformations have many real-world applications, including in the field of particle physics, where they are used to calculate the effects of high speeds on subatomic particles. They are also used in GPS technology, where the effects of special relativity must be taken into account for accurate location measurements.

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