How much work W must be done on a particle with a mass of m to accelerate it from a speed of 0.905c to a speed of 0.980c?
Ok I know that you compute the Lorenz factor gamma but my question is what value would you use for the v? And do you minus anything from it after that?
Subtract the particle energy at 0.905c from the (higher) energy at 0.980c.
They have already told you the v's. When you divide by c in the Lorentz-factor, you will get v/c values of 0.980 and 0.905.
The energy at speeed v is
m c^2 / sqrt[1 - (v/c)^2]
That includes the rest-mass energy.
To calculate the work done on a particle to accelerate it from one velocity to another, we can use the relativistic formula for kinetic energy:
K = (γ - 1) * mc^2
Where:
K is the kinetic energy,
γ (gamma) is the Lorentz factor, and
m is the mass of the particle.
Since we want to find the work done, we need to calculate the change in kinetic energy. The work done can be defined as:
W = ΔK = K final - K initial
Let's break down the steps to solve the problem:
Step 1: Calculate the Lorentz factor (γ)
To calculate the Lorentz factor, we need the velocity of the particle. In this case, the particle is accelerating from a speed of 0.905c to 0.980c.
The speed of light, denoted by c, is approximately 3.00 x 10^8 meters per second. To find the value of v that we should use in the Lorentz factor equation, we need to convert the given velocities to a decimal value of c.
Velocity 1 (initial velocity): v1 = 0.905c
v1 = 0.905 * c = 0.905 * 3.00 x 10^8 m/s
Velocity 2 (final velocity): v2 = 0.980c
v2 = 0.980 * c = 0.980 * 3.00 x 10^8 m/s
Step 2: Use Lorentz factor to compute the change in kinetic energy
Now that we have the values for v1 and v2, we can calculate γ for both velocities using the formula:
γ = 1 / √(1 - v^2 / c^2)
For v1:
γ1 = 1 / √(1 - v1^2 / c^2)
For v2:
γ2 = 1 / √(1 - v2^2 / c^2)
Step 3: Calculate the change in kinetic energy and work done
Finally, we can calculate the change in kinetic energy (ΔK) and the work done (W) using the formula:
ΔK = K final - K initial
W = ΔK
Substitute the values of γ1, γ2 and m (mass of the particle) into the equation to solve for the work done.
Remember to evaluate the signs and units appropriately. The result will be in joules (J) as work is measured in energy units.
Hope this helps!