Question

How much work is required to accelerate a proton from rest up to a speed of \(0.997 c\)

Short answer

Answer: The work required to accelerate the proton is equal to its final kinetic energy, which can be calculated using the relativistic kinetic energy formula. After substituting the given values and performing the calculations, we find that the work required is approximately \(2.34 \times 10^{-10} J\).

Step-by-step solution

Calculate the final kinetic energy

Using the given final speed of the proton (\(v_f = 0.997c\)), the mass of the proton (\(m_p = 1.67 \times 10^{-27} kg\)), and the speed of light (\(c = 3 \times 10^8 m/s\)), we can calculate the final kinetic energy (K_f) using the relativistic kinetic energy formula: \(K_f = (m_pc^2)\Big(\frac{1}{\sqrt{1-\frac{v_f^2}{c^2}}}-1\Big)\) Plug in the values: \(K_f = (1.67 \times 10^{-27} kg \times (3 \times 10^8 m/s)^2 )\Big(\frac{1}{\sqrt{1-\frac{(0.997(3 \times 10^8 m/s))^2}{(3 \times 10^8 m/s)^2}}}-1\Big)\)

Calculate the initial kinetic energy

Since the proton is initially at rest, its initial speed (\(v_i\)) is 0. Using the relativistic kinetic energy formula, we can calculate the initial kinetic energy (K_i): \(K_i = (m_pc^2)\Big(\frac{1}{\sqrt{1-\frac{v_i^2}{c^2}}}-1\Big)\) Plug in the values: \(K_i = (1.67 \times 10^{-27} kg \times (3 \times 10^8 m/s)^2 )\Big(\frac{1}{\sqrt{1-\frac{0^2}{(3 \times 10^8 m/s)^2}}}-1\Big)\) Since the denominator is 1, the entire term inside the parenthesis becomes 0, meaning that \(K_i = 0\).

Calculate the work

Now that we have the final and initial kinetic energies, we can find the work (W) required to accelerate the proton: \(W = K_f - K_i\) Plug in the values: \(W = K_f - 0 = K_f\) So the work needed to accelerate the proton is equal to the final kinetic energy: \(W = K_f\) Lastly, evaluate the final kinetic energy expression to obtain the numerical value for the work required.