How much work W must be done on a particle with a mass of m to accelerate it from rest to a speed of 0.899c ?

Express your answer as a multiple of mc^2 to three significant figures.

To find the work required to accelerate a particle with mass m to a speed of 0.899c, we can use the relativistic kinetic energy equation:

KE = (γ - 1) * mc^2

where γ is the Lorentz factor given by:

γ = 1 / sqrt(1 - (v^2 / c^2))

First, we need to calculate the Lorentz factor γ:

γ = 1 / sqrt(1 - (v^2 / c^2))
= 1 / sqrt(1 - (0.899c)^2 / c^2)
= 1 / sqrt(1 - 0.899^2)
= 1 / sqrt(1 - 0.808201)
= 1 / sqrt(0.191799)
= 1 / 0.438606
≈ 2.2797

Now, we can calculate the work, W:

W = (γ - 1) * mc^2
= (2.2797 - 1) * mc^2
= 1.2797 * mc^2

Finally, the work, W, required to accelerate the particle is approximately 1.280mc^2 to three significant figures.

To calculate the work W required to accelerate a particle from rest to a speed of 0.899c, we need to use the relativistic expression for kinetic energy.

The relativistic kinetic energy is given by:
K = (γ - 1)mc^2

Where:
- K is the kinetic energy of the particle,
- γ is the Lorentz factor, which is defined as γ = 1 / √(1 - (v^2 / c^2)) where v is the velocity of the particle,
- m is the mass of the particle, and
- c is the speed of light in vacuum.

In this case, the particle is initially at rest, so its initial kinetic energy is zero. The final kinetic energy is given by:
K_final = (γ_final - 1)mc^2

To determine the Lorentz factor γ_final, we know that the final speed of the particle is 0.899c. Thus:
v_final = 0.899c
γ_final = 1 / √(1 - (v_final^2 / c^2))

Now, substituting γ_final into the expression for K_final:
K_final = ((1 / √(1 - (v_final^2 / c^2))) - 1)mc^2

Finally, the work W required to accelerate the particle from rest to a speed of 0.899c is equal to the change in kinetic energy:
W = K_final - K_initial
W = ((1 / √(1 - (v_final^2 / c^2))) - 1)mc^2 - 0

By substituting the given values of m (particle mass), c (speed of light), and v_final (final speed), you can calculate the work W as a multiple of mc^2 to three significant figures.