# Motion under a constant force (relativity-related)

1. Oct 13, 2008

### nastassja

1. The problem statement, all variables and given/known data

Use the first order Taylor approximation (1 + $$\epsilon$$)n $$\approx$$ 1 + n$$\epsilon$$ for $$\epsilon$$<<1

to see what the particle's speed v(t) is in the non-relativistic limit of small speeds. The specific approximation you'll want to make is that the dimentionless quantity (Ft/m0c) is much less than 1. Show that for very small values of this quantity, the particle's acceleration is constant, and find out what this acceleration is.

2. Relevant equations

$$\frac{1}{\sqrt{\frac{1}{Ft/m_{0}}+\frac{1}{c^{2}}}}$$

3. The attempt at a solution

I'm not sure what the question is asking. Do I use F=ma, and solve for a using the formula? How do I show that it works with Newtonian mechanics? Also, Do I take a limit as $$\epsilon$$ approaches zero or...what? I don't think I'll have a problem doing the math, I'm just not clear about what is being asked.

Oh, and does epsilon = Ft/mc or just Ft/m? Because Ft/mc isn't in the equation and if I solve for it, the equation may become more complicated to solve for.

2. Oct 13, 2008

### CompuChip

You do precisely what the question says; make an expansion of the square root. So first rewrite your expression to something containing
$$\frac{1}{(1 + \epsilon)^n}$$
where $\epsilon$ is some quantity which you know is small (one is given in the question, big chance that will be your epsilon).

Don't worry about what epsilon will be, you'll see it roll out of the equation. Just focus on getting a form which you can expand.