A proton has momentum with magnitude p0 when its speed is 0.400c. In terms of p0, what is the magnitude of the proton's momentum when its speed is doubled to 0.800c?

Verified step by step guidance

1. Identify the initial conditions and the change in conditions. Initially, the proton's speed is 0.400c and its momentum is p0. The speed is then doubled to 0.800c.
2. Recall the relativistic momentum formula: $p = \gamma m v$, where $m$ is the rest mass of the proton, $v$ is the velocity, and $\gamma$ is the Lorentz factor given by $\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$.
3. Calculate the Lorentz factor $\gamma$ for both speeds. For the initial speed $v = 0.400c$, calculate $\gamma_1 = \frac{1}{\sqrt{1 - (0.400)^2}}$. For the doubled speed $v = 0.800c$, calculate $\gamma_2 = \frac{1}{\sqrt{1 - (0.800)^2}}$.
4. Use the Lorentz factors to find the new momentum at $v = 0.800c$. The new momentum $p_1$ can be expressed as $p_1 = \gamma_2 m v_1$, where $v_1 = 0.800c$. Express $p_1$ in terms of $p_0$ by using the ratio of the Lorentz factors and the change in speed: $p_1 = \frac{\gamma_2}{\gamma_1} \cdot 2p_0$.
5. Simplify the expression to find the momentum $p_1$ in terms of $p_0$ and the calculated values of $\gamma_1$ and $\gamma_2$. This will give the magnitude of the proton's momentum when its speed is doubled to 0.800c.

Key Concepts

Momentum

Momentum is a vector quantity defined as the product of an object's mass and its velocity. In classical mechanics, it is given by the formula p = mv, where p is momentum, m is mass, and v is velocity. However, in relativistic physics, the formula is modified to account for the effects of high speeds approaching the speed of light, leading to the expression p = γmv, where γ (gamma) is the Lorentz factor.

Relativistic Effects

Relativistic effects become significant as an object's speed approaches the speed of light (c). These effects include time dilation, length contraction, and an increase in mass as perceived from an external frame of reference. The Lorentz factor, γ, increases with speed, affecting calculations of momentum and energy for particles moving at relativistic speeds.

Lorentz Factor

The Lorentz factor, denoted as γ, is a crucial component in relativistic physics, defined as γ = 1 / √(1 - v²/c²), where v is the object's velocity and c is the speed of light. This factor quantifies how much time, length, and relativistic mass increase as an object's speed approaches c. It is essential for calculating the momentum of particles at high velocities, as it modifies the classical momentum formula to account for relativistic effects.