Final answer:
Calculating the work to accelerate a particle involves using the relativistic kinetic energy formula and the Lorentz factor. The work is the change in relativistic kinetic energy as the particle accelerates from one speed to another, expressed as a multiple of mc^2.
Step-by-step explanation:
To calculate the amount of work W needed to accelerate a particle with mass m from rest to a speed of 0.890c, we need to apply principles from special relativity since the speed is a significant fraction of the speed of light c. The work done on the particle is the change in its relativistic kinetic energy.
The relativistic kinetic energy KE of a particle is given by:
KE = (\gamma - 1)mc2
where \gamma (gamma) is the Lorentz factor which is 1/sqrt(1 - v2/c2).
To find the work to accelerate from rest to 0.890c:
\gamma = 1/sqrt(1 - (0.890c)2/c2)
Substitute \gamma into the KE equation and calculate the work W as a multiple of mc2.
To calculate the work to accelerate from 0.890c to 0.990c, we perform a similar process but use the difference in kinetic energy between the two speeds.