- How do you calculate proper time (Tau) between two events with coordinates (0s, 0m) and another pair of coordinates in a frame S?
- how much work must be done on a particle of mass m to move it to a speed of 0.905c?
You use the formula:
(Delta S)^2 =
c^2 (Delta T)^2 - (Delta X )^2
Delta S is the distance in space-time between the two events that differ by Delta X in the spatial direction and by Delta T in the time direction.
Now, Delta S is an invariant, i.e. it is the same when you evaluate it in another reference frame. Now, if you evaluate Delta S in the rest frame, then the time difference is the proper time, while the distance in the spatial direction is zero. This means that:
Delta S = c^2 Tau^2
You can compute the wwork done by evaluating the enrgy:
gamma m c^2
When the aprticle was in rest the energy was m c^2, so the difference (i.e. the kinetic energy) is:
(gamma - 1) m c^2