(II) (a) How much work is required to accelerate a proton...

Key Concepts

Relativistic Kinetic Energy

In the framework of special relativity, the kinetic energy of a particle moving at speeds close to the speed of light is given by K = (? - 1)m?c², where m? is the rest mass, c is the speed of light, and ? (the Lorentz Factor) quantifies the deviation from classical mechanics. This expression accounts for the increase in mass-energy as the particle accelerates, differing from the classical ½mv² formulation.

Lorentz Factor

The Lorentz Factor, denoted as ?, is a crucial component in special relativity that describes how time, length, and relativistic mass transform for observers in different inertial frames. It is defined by ? = 1/?(1 - (v/c)²), where v is the object's velocity and c is the speed of light. This factor is essential for calculating both kinetic energy and momentum at relativistic speeds.

Relativistic Momentum

Relativistic momentum extends the classical concept of momentum to account for the effects of special relativity. It is defined as p = ?m?v, incorporating the Lorentz Factor ? to adjust for the increased inertia of an object moving at speeds approaching that of light. This formulation is necessary because classical momentum (mv) is insufficient when velocities are high and relativistic effects become significant.

Transcript

00:02 Our question says that first, a, wants us to calculate how much work is required to accelerate a proton from rest up to a speed, which is a of 0 .99a times the speed of light.

00:16 And then for b, what would be the momentum of this proton? so i also write down here some other useful constant values that we're going to use, and that's that mc squared, which is the rest mass or rest energy of the proton is equal to 930.

00:33 38 .3 mev.

00:34 And then i just went ahead and rearrange that to calculate the mass in units of mev per c squared by dividing by dividing that value by c squared and so that's 938 .3 mep per c squared so first we're going to calculate the work and the work is given by the change in the kinetic energy this is part a, but our initial kinetic energy is zero so the work here is just the final kinetic energy.

00:59 We'll just call this kf which is equal to gamma minus 1 times mc squared.

01:08 So we have that mc squared value written above.

01:12 It's 938 .3 mev, and gamma is one divided by the square root of 1 minus v squared over c squared.

01:22 Well, v squared is 0 .998c, and that's going to, or v is 0 .998c, so v squared is 0 .9 squared.

01:31 So the c squared are going to cancel and you're going to have 0 .998c.

01:36 8, square root extend to include all that.

01:43 And this is minus 1 multiplied by mc squared, which is 938 .3m .v.

01:57 Okay.

02:23 Gv.

02:32 Okay, we can box that in as our solution to part a.

02:35 Now for part b, we're asked to find the momentum...