The amount of work required to accelerate a proton from rest to a speed of 0.66c is approximately 3.15 x 10^(-13) joules.
To calculate the work required to accelerate a proton from rest to a speed of 0.66 times the speed of light (c), we can use the relativistic kinetic energy formula. The kinetic energy (KE) is given by KE=(γ−1)mc^2, where γ is the Lorentz factor and m is the rest mass of the proton.
The Lorentz factor is defined as γ=
, where v is the velocity of the proton.
Given that the proton is accelerated to a speed of 0.66c, we can substitute this velocity into the equations. Firstly, find γ, and then use it to calculate the kinetic energy. The work (W) done is equal to the change in kinetic energy.
γ≈1.36
Now, use the relativistic kinetic energy formula:
KE=(γ−1)mc^2
KE≈(1.36−1)m(299,792,458m/s)^2
KE≈0.36mc^2
This represents the kinetic energy gained by the proton. The work done to achieve this kinetic energy is equal to the change in kinetic energy, so W=0.36mc^2. The mass of the proton (m) is approximately 1.67×10^−27 kg.
W≈0.36×(1.67×10^−27)×(299,792,458)^2
W≈3.15×10^−13 joules
Therefore, the amount of work required to accelerate the proton is approximately 3.15×10^−13 joules.