# Constantly accelerating rocket algebra problem

1. Mar 21, 2016

### Fek

1. The problem statement, all variables and given/known data
• Rocket is accelerating constantly. Let S' be instantaneous rest frame of rocket and S be frame in which rocket is observed moving at velocity v.

2. Relevant equations
Given: $$dv = dv' (1 - v^2)$$

Must prove:
$$\frac{dv}{dt} = \frac{dv'}{dt'} (1 - v^2)^{\frac{3}{2}}$$

3. The attempt at a solution

So differentiate given equation with respect to t and use chain rule to get in terms of t'

$$\frac{dv}{dt} = \frac{dt'}{dt} * \frac{d}{dt'}[dv'(1 - v^2)]$$
We also know
$$\frac{dt'}{dt} = (1 - v^2)^{\frac{1}{2}}$$
as t' is proper time.

We also have:
$$\frac{d}{dt'} (dv' (1 - v^2) = \frac{dv'}{dt'} (1 - v^2)$$

We have the answer! Except this final step isn't right because v is a function of t' as well and chain rule must be used?

2. Mar 21, 2016

### PeroK

If you have a functional equation, you can differentiate it. If you have an equation involving infinitesimal differentials, you can't differentiate it. Instead, you can divide by another infinitesimal differential.