Let the 4-velocity be
with the constraint
By Newton's law
Also proper time is related to coordinate time by
where
.
Put everything together
which gives
The other equation we have is
The calculation is a little complicated
Clearly, this relation can't be inverted.
A free particle, having rest mass Mo, is in vacuum and initially at rest in the lab frame S. It undergoes an accele...
A free particle, having rest mass Mo, is in vacuum and initially at rest in the lab frame S. It undergoes an acceleration under the action of a constant pure force F = (fo, fr, 0.0), where 0 c dt Find its 4-velocity U as a function of time and force. Sketch the graphs of the dependence of normalised 3-velocity 3 v/c and the Lorentz factor y(t) of the particle as a function of time t
A particle of rest mass mo and charge q is accelerated from rest by a uniform (in the lab frame) electric field Ei. What are the velocity and position of the particle (as a function of time) a. in the lab frame? b. in the rest frame of an observer moving with a velocity vok relative to the lab? c. (Optional) Plot the position and speed of an electron in a uniform field of magnitude 1 MV/m for the time...
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A particle with mass 4.0 kg is initially at rest at the origin (x=0). It then moves along the x-axis acted on by a force F(x)=18.0-2.0x2, where F is in Newtons if x is expressed in meters. Find the kinetic energy of the particle when it reaches x=3 m. A Honda Civic (mass=1090 kg) is initially traveling due East at 18.3 m/s when it suddenly collides head on with a Toyota Camry (mass=1612 kg) initially traveling due West at 31.8...
4. Field transformations: In the lab frame E 4 V/m, B -3y T, and a point charge q = 1C is observed to be moving with velocity v 2 m/s. a) What is the electric field E' measured in the frame of reference of q? Determine E' by ensuring that the Lorentz force applied on charge q is identical in both reference frames. b) Is this charge being accelerated or not under the influence of fields E and B? Discuss...
A small rocket having an initial weight of 3000 lb (including 2400 lb of fuel), and initially at rest, is launched vertically upward. The rocket burns fuel at a constant rate of 80 lbs, which provides a constant thrust, T, of 8000 lb. The instantaneous weight of the rocket is w(t) = 3000 - 80t lb. The drag force, D, experienced by the rocket is given by D = 0.005g(dy/dt)^2 lb, where y is distance in ft, and g =...