Final answer:
The amount of work required to accelerate the proton to a speed of 0.75c can be calculated as 0.93 Joules.
Step-by-step explanation:
To calculate the amount of work required to accelerate a proton to a speed of 0.75c in a particle accelerator, we can use the formula:
Work = change in kinetic energy
The change in kinetic energy can be calculated as:
Change in kinetic energy = final kinetic energy - initial kinetic energy
The initial kinetic energy of the proton is given by:
Initial kinetic energy = (1/2) * mass * initial velocity^2
Plugging in the values provided:
Initial kinetic energy = (1/2) * (1.67 × 10^-27 kg) * (0)^2 = 0
Since the proton starts from rest, its initial kinetic energy is 0.
The final kinetic energy can be calculated as:
Final kinetic energy = (1/2) * mass * final velocity^2
Plugging in the values provided:
Final kinetic energy = (1/2) * (1.67 × 10^-27 kg) * (0.75c)^2
Using the fact that c = 3.00 × 10^8 m/s, we can convert the final velocity:
Final velocity = 0.75c = 0.75 * (3.00 × 10^8 m/s) = 2.25 × 10^8 m/s
Plugging in the values:
Final kinetic energy = (1/2) * (1.67 × 10^-27 kg) * (2.25 × 10^8 m/s)^2
Simplifying the equation:
Final kinetic energy = 0.93 Joules
The amount of work required to accelerate the proton is equal to the change in kinetic energy:
Work = Final kinetic energy - Initial kinetic energy = 0.93 Joules - 0 Joules = 0.93 Joules