Question

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

Short answer

Answer: The amount of work required to accelerate a proton from rest to a speed of 0.997c is approximately \(3.209 \times 10^{-13} J\).

Step-by-step solution

Calculate the Lorentz factor

To calculate the Lorentz factor, use the formula: \(\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\) Plug in the given speed, \(v = 0.997c\), and solve for \(\gamma\): \(\gamma = \frac{1}{\sqrt{1 - (0.997)^2}} \approx 7.088\)

Calculate the final relativistic kinetic energy

Now that we have the Lorentz factor, we can calculate the final relativistic kinetic energy using the formula mentioned earlier: \(K = (\gamma - 1)mc^2\) Use the mass of a proton, \(m \approx 1.67 \times 10^{-27} kg\), the speed of light \(c = 3.00 \times 10^8 m/s\), and the calculated Lorentz factor \(\gamma \approx 7.088\): \(K = (7.088 - 1)(1.67 \times 10^{-27} kg)(3.00 \times 10^8 m/s)^2 \approx 3.209 \times 10^{-13} J\)

Calculate the work required

Since the proton starts at rest, its initial kinetic energy is zero. Therefore, the work required to accelerate the proton to the given speed is equal to its final relativistic kinetic energy: \(W = K_f - K_i = 3.209 \times 10^{-13} J\) So, the amount of work required to accelerate a proton from rest to a speed of 0.997c is approximately \(3.209 \times 10^{-13} J\).

Key concepts

Lorentz Factor

The Lorentz factor is a key element in understanding motion at speeds approaching the speed of light, as described in special relativity. It is denoted by the symbol \(\gamma\) and is defined by the equation \(\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\), where \(v\) is the velocity of an object and \(c\) is the speed of light in a vacuum. The Lorentz factor measures the amount of time dilation, length contraction, and increase in mass that occurs as the speed of an object approaches the speed of light. As an object speeds up, the Lorentz factor increases, causing ordinary Newtonian mechanics to give way to relativistic effects.

The use of the Lorentz factor in calculating relativistic kinetic energy is crucial, especially when dealing with particles such as protons in accelerators. For a proton to reach near-light speeds, a significant amount of work must be done, which includes overcoming the effects encapsulated by the Lorentz factor. As demonstrated in the exercise, the proton's speed at 0.997 times the speed of light leads to a Lorentz factor of approximately 7.088, indicating that relativistic effects are significant.

Special Relativity

Special relativity, pioneered by Albert Einstein in 1905, fundamentally changed our understanding of space, time, and energy. The theory includes two core postulates: the laws of physics are the same in all inertial frames of reference, and the speed of light in a vacuum is constant for all observers, regardless of their relative motion. Special relativity introduces novel concepts such as time dilation (time moving slower for a moving object than for a stationary observer) and length contraction (an object moving at high speed appears shorter along the direction of motion from the viewpoint of a stationary observer).

When applying special relativity, such as in the exercise on accelerating a proton to near-light speeds, we encounter scenarios where velocities are substantial fractions of the speed of light \(c\). In these contexts, the relativistic formulation of kinetic energy \(K = (\gamma - 1)mc^2\) replaces classical expressions, reflecting the fact that increasing an object's speed by even small amounts as it nears light speed demands vastly more energy. This formulation showcases one of special relativity's most famous equations, \(E=mc^2\), revealing the profound link between mass and energy.

Work-Energy Theorem

The work-energy theorem is a principle that relates the work done on an object to the change in its kinetic energy. In classical mechanics, it states that the work done by the net force acting on an object equals the change in its kinetic energy. The theorem is given by the equation \(W = \Delta K = K_f - K_i\), where \(W\) is work, \(\Delta K\) is the change in kinetic energy, \(K_f\) is the final kinetic energy, and \(K_i\) is the initial kinetic energy.

In the context of special relativity, the theorem still holds but requires the relativistic form of kinetic energy. As seen in the example problem, the work required to accelerate a proton from rest to 0.997c is calculated using the final relativistic kinetic energy. In relativistic terms, the kinetic energy of any object includes the mass-energy equivalent as per Einstein's theory, representing yet another layer of complexity in understanding how forces and energy interact at high speeds. Specifically, the energy required grows exponentially as the particle speed approaches the speed of light, differing vastly from non-relativistic scenarios where energy and speed have a quadratic relationship.