Electron accelerated from .89c to .97c

[danielatha4]

Homework Statement

SLAC, the Stanford Linear Accelerator Collider, located at Stanford University in Palo Alto, California, accelerates electrons through a vacuum tube two miles long (it can be seen from an overpass of the Junipero Serra freeway that goes right over the accelerator). Electrons which are initially at rest are subjected to a continuous force of 1.2×10-12 Newton along the entire length of two miles (one mile is 1.6 kilometers) and reach speeds very near the speed of light.

Determine how much time is required to increase the electrons' speed from 0.89c to 0.97c.

Approximately how far does the electron go in this time? (What is approximate about your result?)

Homework Equations

The Attempt at a Solution

I figured the acceleration of the electron would be equal to the force, 1.22x10^-12, divided by the mass of an electron, 9.1x10^-31. Then I used V_f=V_o+at to solve for time. However, this was not correct.

 

[Maxim Zh]

It's a relativistic problem and you have to use relativistic equations.

The Newton's second law is correct here in the momentum form:

$$\frac{\partial \mathbf{p}}{\partial t} = \mathbf{F}.$$

The speed dependence of momentum is nonlinear:

$$\mathbf{p} = m\gamma(v)\mathbf{v}.$$

The equation for x(t) will be very complicated. Perhaps the ultrarelativistic approximation will simplify it. But I'm not sure it could be used for v=0.89c.

 

[danielatha4]

My teacher did give us the equation $$\Delta$$P=F_net$$\Delta$$t

However, solving the final and initial relativistic momentum for the change in momentum, and dividing by the force applied gives me 4.64*10^-12 seconds, and that's not right.

 

[Maxim Zh]

My result is 4.64*10^-10 s.