# Power of the mass driver

Yesterday I added power management mechanics. For tests I put most of the power consumption figures from top of my head. But this is going to be hard sci-fi, why not check it exactly?

Let’s start with assumptions. Mass driver can apply thrust of, on average, 300kN to a 10kg slug in 0.15s. It needs 0.35s to load another one, but that is not relevant here. Can I get power requirements from that? I should. I assumed efficiency of 90%, which seems sensible for an electromagnetic mass driver.

Electrical energy required: $E_e = P*t$

Kinetic energy of $E_k = { m * v^2 \over 2 }$

Our efficiency is 90%, so $E_k = 0.9 * E_e$

From that I can get actual equation: ${ m * v^2 \over 2 } = 0.9*P*t$

I will also need to get velocity from acceleration $v = a*t$ and acceleration from force $F=m*a$ so $a={F \over m}$.  So let’s do actual math:

$P = { m*v^2 \over 2*t*0.9 } = { m * (a*t)^2 \over 1.8 *t } = { m * a^2 * t^2 \over 1.8 * t } = { m * a^2 * t \over 1.8 } = { m * ( { F \over m } )^2 * t \over 1.8} = { m * { F^2 \over m^2 } * t \over 1.8} = { { F^2 \over m } * t \over 1.8} = { F^2 * t \over 1.8 * m }$

Unit check:

${ N^2 * s \over kg } = { ({ kg * m \over s^2 })^2 * s \over kg } = { kg^2 * m^2 \over s^4 } * { s \over kg } = { kg * m^2 \over s^3 } = W$

Okay, so we have an equation. Let’s put in some data:

$P = { F^2 * t \over 1.8 * m } = {300kN^2 * 0.15s \over 1.8 * 10kg } = 0.75GW$

Whoa. 750MW is quite a lot, but not out of scope of nuclear reactor. Since this power is needed only for 0.15s, it’s likely that there are ultracapacitors to handle this peak.

But wait, there’s more. Physics is fun and answers lot more questions.

#### How fast is the slug?

This one is easy: $v=a*t; a={f \over m}; v={f*t\over m } = { 300kN * 0.15s \over 10kg } = 4.5 [{ km \over s }]$

#### How hot is it?

Our efficiency is 90%, so it’s 10% of 750MW over 0.15s, which is 11.25MJ. Specific heat of steel is $475 { J \over kg*K }$, so the slug should have roughly 2500K, give or take initial temperature. That means that slug should be glowing red.

Which, fortunately, it does.