Given: Two particles are created in a high-energy accelerator and move off in opposite directions. The speed of one particle, as measured in the laboratory, is 0.580c, and the speed of each particle relative to the other is 0.920c.
Problem:What is the speed of the second particle, as measured in the laboratory?
I'm not even sure where to start...
In the frame of reference of the slower particle (as seen in the lab) the speed of the other particle is +0.920c. To get the lab speed of the faster particle, you have to do a transformation back to lab coordinates, with a relative velocity of 0.580c
Th relativistic formula for adding relative velocities in this way is
V' = (V1 + V2)/[1 + V1*V2/c^2]
= 1.500 c/[1.5336] = 0.978c
I tried this and it said it was incorrect. Is there another way?
To solve this problem, we can use the concept of relativistic velocity addition. In special relativity, velocities do not simply add up in the same way as in classical mechanics. Instead, we need to use a formula that takes into account the relativistic effects.
To get started, let's define the following variables:
- vA: Speed of the first particle as measured in the laboratory (0.580c)
- vB: Speed of the second particle as measured in the laboratory (unknown)
- vAB: Relative speed of the second particle with respect to the first (-0.920c)
Now, we can use the relativistic velocity addition formula to find the speed of the second particle in the laboratory frame. The formula is as follows:
vA + vB / (1 + (vA * vB) / c^2)
Plugging in the given values, we have:
0.580c + vB / (1 + (0.580c * vB) / c^2) = -0.920c
Simplifying the equation, we get:
(0.580c + vB) / (1 + 0.580vB / c) = -0.920c
Cross-multiplying, we get:
0.580c + vB = -1.064c + (-0.920vB)
Combining like terms, we have:
1.500vB = -1.644c
Dividing both sides by 1.500, we get:
vB = -1.096c
Since speed cannot be negative in this context, we must take the absolute value:
vB = 1.096c
Therefore, the speed of the second particle, as measured in the laboratory, is 1.096 times the speed of light (c).