The velocity of electron near speed of light?

[randybryan]

This isn't a homework question, simply one I found in a book that I'm trying to do:

momentum p, of electron at speed v near speed of light increases according to formula

p = $$\frac{mv}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}$$

if an electron is subject to constant force F, Newton's second law of describing motion is

$$\frac{dp}{dt}$$ = $$\frac{d}{dt}$$ $$\frac{mv}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}$$ = F

This all makes sense to me. It then says, find v(t) and show that v --> c as t --> infinity. Find the distance traveled by the electron in time t if it starts from rest.

Now I could get an expression for v by using the first formula, but I don't understand how I can show that v -->c as t --> infinity as t isn't in the equation. I haven't even attempted the second part, but I'm assuming some integration is involved

Can anyone help?

 

[jeppetrost]

What you want to do to find v(t) is to solve the differential equation:
$$F = \frac{d}{dt} \left( m v(t) \gamma (t)\right)$$
for v(t).
Now, you can differentiate the product to get
$$F = m a(t) \gamma (t) + m v(t) \left( \frac{v(t)a(t)}{c^2}\gamma^3(t)\right)=m a(t) (\gamma + \beta ^2(t)\gamma ^3(t)), ~~~~ \beta=\frac{v(t)}{c}$$
Now, solving this (I used Maple) and imposing the condition v(0)=0, one gets the expression
$$v(t)=\frac{Fct}{\sqrt{F^2 t^2 + c^2 m^2}}$$
As you can see, v approaches c a time goes to infinity.
The expression is easily (with some computer) integrated to give you an expression for the distance traveled over time:
$$d(t)=\frac{c}{F} \sqrt{t^2F^2+c^2m^2}$$

 

[starthaus]

This isn't a homework question, simply one I found in a book that I'm trying to do:

momentum p, of electron at speed v near speed of light increases according to formula

p = $$\frac{mv}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}$$

if an electron is subject to constant force F, Newton's second law of describing motion is

$$\frac{dp}{dt}$$ = $$\frac{d}{dt}$$ $$\frac{mv}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}$$ = F

This all makes sense to me. It then says, find v(t) and show that v --> c as t --> infinity. Find the distance traveled by the electron in time t if it starts from rest.

Now I could get an expression for v by using the first formula, but I don't understand how I can show that v -->c as t --> infinity as t isn't in the equation. I haven't even attempted the second part, but I'm assuming some integration is involved

Can anyone help?

F=m*d/dt(v/sqrt(1-(v/c)^2)

so,

F/m=d/dt(v/sqrt(1-(v/c)^2)


Since F/m is constant, the above becomes a very simple differential equation with the solution:

v=at/sqrt(1+(at/c)^2)

For at<<c, you recover the Newtonian equation v=at

If you integrate one more time, you will get x as a function of a and t. Indeed:

dx/dt=at/sqrt(1+(at/c)^2)


x(t)=c^2/a*(sqrt(1+(at/c)^2)-1)

Again, for at<<c, you recover the Newtonian formula x(t)=at^2/2