Special Relativity - addition of velocities

My answer comes to 2ct_0 - \frac{2lv}{c^2} In summary, the conversation is about a K-meson decaying into two pi-mesons, and the question of what is the greatest speed one of the pi-mesons can have if the K-meson is traveling at a speed of 0.9c. The equations used are u\prime = \frac{u-v}{1-\frac{v}{c^2}u} and \frac{u+v}{1+\frac{uv}{c^2}}. The conversation then moves on to a problem involving a pulsed radar source and a moving meteorite, with the question of evaluating the time interval between
  • #1
Brewer
212
0

Homework Statement


A K-meson at rest decays into two pi-mesons, and each pi meson has a speed of 0.85c.

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


Homework Equations


[tex]u\prime = \frac{u-v}{1-\frac{v}{c^2}u}[/tex]


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.
 
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  • #2
The right-hand side of the addition of velocities formula should be

[tex]\frac{u+v}{1+\frac{uv}{c^2}}[/tex]
 
  • #3
With the numbers I have for this question that gives a velocity of 1.35c. Obviously not right.
 
  • #4
Do it by hand. Then learn how to use your calculator.
 
  • #5
Fredrik said:
The right-hand side of the addition of velocities formula should be

[tex]\frac{u+v}{1+\frac{uv}{c^2}}[/tex]

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

With u= .9c, v= .85c you should have
[tex]\frac{(.9+ .85)c}{1+ (.9)(.85)}[/itex]
That is NOT larger than c!
 
  • #6
I have done - I think I may have forgotten brackets or something. I get an answer 0.991c now.
 
  • #7
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[/tex] (with [tex]t_0 < \frac{l}{c}[/tex]

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
I think that I've done this last bit.
 

1. What is the concept of "addition of velocities" in special relativity?

The addition of velocities in special relativity refers to the mathematical formula used to calculate the combined velocity of two objects moving at different speeds. This formula takes into account the principles of relativity, which state that the laws of physics should be the same for all observers moving at constant speeds.

2. How does special relativity explain the limitation of the speed of light?

Special relativity proposes that the speed of light is the ultimate speed limit in the universe, and nothing can travel faster than it. This is because as an object approaches the speed of light, its mass and energy increase infinitely, making it impossible to accelerate to or surpass the speed of light.

3. Can an object with mass ever reach the speed of light?

No, according to the principles of special relativity, an object with mass can never reach the speed of light. As an object's speed increases, its mass also increases, making it more and more difficult to accelerate. At the speed of light, an object's mass would become infinite, making it impossible to accelerate any further.

4. How does special relativity affect the concept of time and space?

Special relativity introduces the concept of spacetime, which combines the three dimensions of space with the dimension of time. It proposes that time and space are relative and can be affected by an observer's speed and gravitational forces. This leads to phenomena such as time dilation and length contraction, where time and space appear to be different for observers moving at different speeds.

5. What are the implications of special relativity in daily life?

Special relativity has many practical applications in modern technology, such as GPS navigation systems and particle accelerators. It also helps us understand the behavior of objects at high speeds and in extreme conditions, such as near the speed of light or in the presence of strong gravitational forces. Additionally, it has led to important insights in the fields of cosmology and quantum mechanics.

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