Special relativity: Simple Lorentz transformation question

In summary, the space-cruiser sees the fire-engine leave its station 6363 m due north from Cape Canaveral, 10^-5 s earlier, and the shuttle launch occurring after the fire-engine leaves the station. The space-cruiser's navigator calculates the distance between the two events to be 30000 m.
  • #1
bojo
5
0

Homework Statement



Observer O sees a fire-engine leave its station 6363 m due
north from Cape Canaveral, where the super-shuttle Lorentz had been
launched 10^-5 s earlier. A space-cruiser flying north-east sees these two events
also 10^-5 s apart, but with the shuttle launch occurring after the fire-engine
leaves the station.

(a) Show that the speed of the space-cruiser relative to the Earth is 12c/13.
[5 marks]

(b) How far apart does the navigator on the space-cruiser measure the two
events to be? [3 marks]

Homework Equations



[tex]
\Delta t ' = \gamma(\Delta t - v\Delta x/c^2)
[/tex]

The Attempt at a Solution



I know this must be quite simple, I'm just not quite sure where to start. I've fiddled with the Lorentz transformation equations to no avail - i think its the time interval concept which confuses me here (what actually is the interval? 10^-5 s?)

I tried putting in the value of the speed given (*cos45 as it's heading north-east) to try obtain the times given or something related to them but also this was pretty useless.

I imagine part b is quite trivial once part a is complete

Helpfull shoves most appreciated,
Ben
 
Last edited:
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  • #2
The equation for the Lorentz transformation you have there is for when the second reference frame moves with speed v in the +x direction and the spatial axes are parallel to each other. You need to orient your coordinate systems so that these conditions are met. Once you do that, figure out the coordinates of the two events in one reference frame and then use the transformation to find the interval in the second frame.
 
  • #3
i understand the need to have the parallel spatial axes but i don't quite understand how to figure out the coordinates?

is the equation i stated the correct one to use in this example?
 
  • #4
Align the x-axis with the NE direction and the y-axis in the NW direction. Place Cape Canaveral at the origin, so its (x,y) coordinates are (0,0). The fire station is 6363 m due north of that point. Use plain old trigonometry to figure out what the (x,y) coordinates of the station would be. (There's no relativity involved yet.)
 
  • #5
vela said:
Align the x-axis with the NE direction and the y-axis in the NW direction. Place Cape Canaveral at the origin, so its (x,y) coordinates are (0,0). The fire station is 6363 m due north of that point. Use plain old trigonometry to figure out what the (x,y) coordinates of the station would be. (There's no relativity involved yet.)

AH! this is where I've been going wrong. Stupidly I've been just using the north component of the spaceship's velocity - makes no sense to do that now you've pointed it out.

Thanks very much!
 

1. What is special relativity?

Special relativity is a theory developed by Albert Einstein that describes the relationship between space and time. It states that the laws of physics are the same for all observers in uniform motion, and that the speed of light is constant regardless of the observer's frame of reference.

2. What is the Lorentz transformation?

The Lorentz transformation is a mathematical formula used in special relativity to describe how measurements of space and time change for different observers in relative motion. It takes into account the effects of time dilation and length contraction.

3. How does the Lorentz transformation work?

The Lorentz transformation uses the speed of light (c) as a constant and applies it to the equations for time (t) and space (x, y, z) to account for the differences in measurements between observers in relative motion. It also takes into account the concept of simultaneity, where two events that are simultaneous for one observer may not be simultaneous for another.

4. What is the significance of the Lorentz transformation in special relativity?

The Lorentz transformation is integral to the theory of special relativity as it allows for the consistency of physical laws across different frames of reference. It also explains the observed phenomena of time dilation and length contraction, which have been confirmed through experiments such as the famous Michelson-Morley experiment.

5. Can the Lorentz transformation be applied to all situations?

No, the Lorentz transformation is only applicable in situations where objects are moving at constant speeds and in straight lines. It does not take into account the effects of acceleration or gravity, which are better described by the theory of general relativity.

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