# Minimum energy to accelerate a mass?

1. May 19, 2015

### jerromyjon

I have a similar logical "gap" in my understanding that I still haven't resolved... which seems to be right on point with this thread if I call it "minimum amount of energy required to accelerate a given mass".

From what I know about SR, accelerating mass takes energy which increases exponentially as this mass approaches c. I just don't know at what speed this effect becomes measurable...

2. May 20, 2015

### Ibix

It isn't an exponential growth. The energy of a massive particle moving at velocity v is $E=\gamma mc^2$, where $\gamma=(1-v^2/c^2)^{-1/2}$. If you Taylor expand the expression for $\gamma$ you get
$$E=mc^2 +\frac{1}{2}mv^2 +\frac{3}{4}m \frac{v^4}{c^2} +...$$The first term is mass energy. The second is Newtonian kinetic energy. The third and later terms are where relativity disagrees with Newton on energy. So the effect is measurable when that term is measurable.

I don't know if there's an answer to your question in practical terms since I'm not current on the precision of energy measurement devices, and that probably depends on your application anyway. But that's the theoretical basis for an answer.

3. May 20, 2015

### jerromyjon

Thank you, I appreciate your response! I'm still trying to learn what is still "greek" to me, but that funky y that you all call gamma... isn't that only zero at zero velocity?

4. May 20, 2015

### Staff: Mentor

What do you get when you plug v = 0 into the formula for gamma? $$\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$$

5. May 20, 2015

### jerromyjon

Ummm, If I set c=1 then its 1 but I'm not even sure about that c=1 scenario.

6. May 20, 2015

### jerromyjon

oh duh nevermind gamma has to be 1 to be at rest energy..

7. May 20, 2015

### jerromyjon

Ok. So what I meant was gamma is 1 at rest energy, which would only be at rest velocity. Suppose we do the M&M set-up with "identical" rockets in opposite directions, along with an identical complete system at a known relative velocity parallel to the launch vectors. This is where I'm unsure how to predict how this system evolves, but now I recall something about the relativistic "tether" problem, where the distance changes, severely complicating this visually. I'm going to play with that interactive Minkowski diagram (Thanks Ibix!) and see if I can figure it out...