# Special Relativity - addition of velocities

1. Apr 28, 2007

### Brewer

1. The problem statement, all variables and given/known data
A K-meson at rest decays into two pi-mesons, and each pi meson has a speed of 0.85c.

If a K-meson travelling at a speed of 0.9c decays, what is the greatest speed the one of the pi-mesons can have?

2. Relevant equations
$$u\prime = \frac{u-v}{1-\frac{v}{c^2}u}$$

3. The attempt at a solution
After plugging in v=0.9c and u=0.85c I get 3 different answers (3 different derivations of the above equation)

Initially I got 1.11c which is clearly wrong, I think that answer came from just not cancelling properly and putting things in my calculator wrong.

Next I got a value of -0.213c from the above equation that I just copied from my notes.

Finally I got a value -0.213c from a similar equation to that above, but with all the signs on the RHS switched, when I tried to derive the equation. (I didn't actually know that this was the same answer - I've only just worked out the fractional value while typing this!). I am however confused, as surely a complete change of sign throughout the equation would change the sign of the answer. So why have I got the same answer?

Any help at all would be gratefully received - my main problem with Special Relativity is deciding which equation to use for the data given.

2. Apr 28, 2007

### Fredrik

Staff Emeritus
The right-hand side of the addition of velocities formula should be

$$\frac{u+v}{1+\frac{uv}{c^2}}$$

3. Apr 29, 2007

### Brewer

With the numbers I have for this question that gives a velocity of 1.35c. Obviously not right.

4. Apr 29, 2007

### Meir Achuz

Do it by hand. Then learn how to use your calculator.

5. Apr 29, 2007

### HallsofIvy

Staff Emeritus
With u= .9c, v= .85c you should have
$$\frac{(.9+ .85)c}{1+ (.9)(.85)}[/itex] That is NOT larger than c! 6. Apr 29, 2007 ### Brewer I have done - I think I may have forgotten brackets or something. I get an answer 0.991c now. 7. Apr 29, 2007 ### Brewer On a similar note, could you explain my next question to me please. I can't work out the variables or equations to use (my main problem with special relativity). A pulsed radar source is at rest at the point x=0. A large meteorite moves with constant velocity v towards the source, and is at the point x=-l at t=0. A radar pulse is emitted by the source at t=0, and a second pulse at [tex]t=t_0$$ (with $$t_0 < \frac{l}{c}$$

The pulses are reflected by the meteorite and returned to the source.

i)On a 4-space diagram draw the paths (world lines) of the source, the meteorite and the outgoing and reflected pulses.
ii) Evaluate the time interval between the arrivals at x=0 of the two reflected radar pulses.

Although I've never been taught about world lines (at least I can't see them in my notes and I don't remember them) I think I've done the first part. The line for the meteorite is a straight line beginning at x=-l and continuing onwards. The source is a straight vertical line (i.e. stays at x=0, but continues along the ct axis). The first pulse starts at the origin and continues in a straight line (steeper gradient than that of the rock) until it hits the line representing the rock, at which point it will be the negative gradient of before until it hits the ct axis once more. The second pulse will be the same except that it starts higher on the ct axis, will hit the rock first and will return to the detector first.

Now for the second part I'm really confused. I can't work out how to tackle this problem. I would assume that my rest frame would be that of the source, and my movement frame will be that of the rock (i.e the frame moves with velocity v through the rest frame).

Other than that I can't work out any of the variables to use. I think that x=0, but I'm not sure. I also can't tell what my target variable is (t'?)

If you have any pointers I would be appreciative.

8. Apr 29, 2007

### Brewer

I think that I've done this last bit.