a) How much work must be done on a particle with a mass of m to accelerate it from rest to a speed of 8.4×10−2 c ? (Express the answer in terms of mc^2.)
b)How much work must be done on a particle with a mass of m to accelerate it from a speed of 0.900c to a speed 0.984 c ? (Express the answer in terms of mc^2.)
To find the work done on a particle, we can use the relativistic equation for kinetic energy:
K = (γ - 1)mc^2
where K is the kinetic energy, γ is the Lorentz factor given by γ = 1/√(1 - (v^2/c^2)), m is the mass of the particle, and c is the speed of light.
a) For a particle accelerated from rest to a speed of 8.4×10^-2 c, we need to find the Lorentz factor γ at this speed. Plugging in the values into the equation, we have:
v = 8.4×10^-2 c
γ = 1/√(1 - (8.4×10^-2)^2)
To simplify the calculation, let's use the approximation that (8.4×10^-2)^2 ≈ 7×10^-3. Now, substitute this into the equation:
γ = 1/√(1 - 7×10^-3)
To get the kinetic energy, we can use the equation K = (γ - 1)mc^2. Since the particle starts from rest, the initial kinetic energy is zero. Hence, the work done is equal to the change in kinetic energy, which is just K:
Work = K = (γ - 1)mc^2
Substituting the value of γ into the equation, we have:
Work = [(1/√(1 - 7×10^-3)) - 1]mc^2
This expression gives the amount of work that needs to be done on the particle, expressed in terms of mc^2.
b) For a particle accelerated from a speed of 0.900c to a speed of 0.984c, we can follow the same approach as before. First, we find the Lorentz factor γ at the final speed:
v = 0.984c
γ = 1/√(1 - (0.984)^2)
Using the approximation that (0.984)^2 ≈ 0.968, we substitute into the equation:
γ = 1/√(1 - 0.968)
The work done is again given by the change in kinetic energy, which is K:
Work = K = (γ - 1)mc^2
Substituting the value of γ into the equation, we obtain:
Work = [(1/√(1 - 0.968)) - 1]mc^2
This expression gives the amount of work that needs to be done on the particle, expressed in terms of mc^2.