# Running in Space

Silly question here with a long back story, but basically: how fast can you accelerate by running in space? A few very naive assumptions: You have a cross sectional area of about 1/3 m^2, and the density of space is about 5 protons/m^3, so you have a mean free path of about half a meter, or your average stride length anyway. Say you could step on every proton in a way that applies the entire force of your step opposite to your direction of motion.

Assuming you can dodge all protons that would impede your motion, and that they're all at rest, how fast would you accelerate?

Values used/assumed:
mass of proton = 1.67*10^-27 kg
mass of person = 68 kg
Force of step = 100 N
Time duration of step = 0.01 s

Using the impulse-momentum equation here: https://www.physicsforums.com/showthread.php?t=440335 for getting a new velocity from a force, and then numerically integrating over time, I get the horrible result that you would have moved 14km in 15 minutes of running.

That is, I'm saying:
$v[t+1]=v[t]+F*dt/m_{you}$
Which doesn't include the mass of the proton, which it seems like it should, so I feel like I can't assume that force. It was taken from typical earth equations for kicking things, but I've seen that ion drives have forces in milliNewtons.

So for any of you bored and willing to appease my awful assumptions for the sake of curiosity, what am I doing wrong?

sorry...but you are not going anywhere!

say on earth you have a walkway of stepping stones spaced out at 0.5 meters..the only reason why you can run on them is because you can actually push (fall) yourself from one of them to the point where you can reach the next one...since the mass of the "stepping stone" is much larger than yours

on space, you will not be able to 'fall' from one proton to the next one, because there is no gravity.

let's go to water now

say you are floating on water but your motion does not propel from water itself and you have ping pong balls spaced out at 0.5 meters along the surface of the water...can you pull yourself from one of them enough (0.5 m) so you can reach the next one? I don't think so...you will simply end up moving the ping pong to the other side of you, the center of mass of you and the ping pong will stay in the same place and presumably you will move up a tiny amount of distance ....and that's it...you will not be in a position to reach the very next ping pong and so you are stuck.

BruceW
Homework Helper
vityav, your equation is correct. One problem pointed out by gsal is that when the person pushes off from the first few (5 or so) protons which are within his reach, it will take a while before he is within reach of the next proton. This is only a problem when he is starting from rest. Once he gets some speed, he should pass by enough protons.

You said that it was a horrible result. I don't know your method of integration, but your result shouldn't be surprising, since the force of each step is 100N and time duration 0.01s, these are everyday numbers, so we should expect an everyday answer, which we do get (kilometres is an everyday number, although people don't run this fast).

Your equation could apply to pushing off of anything, like tennis balls instead of protons, because you have specified the impulse gained by pushing off from them, so information about the object (like its mass) doesn't enter into the equation.

The reason ion drives have less force is because they do not give such a huge impulse to the particles they spew out. Your equation is saying that the person gives each proton an impulse of 1Ns, this is equivalent to a particle accelerator giving the proton roughly 10^15 TEV, which is far beyond current technology.

D H
Staff Emeritus
what am I doing wrong?
You are ignoring what happens to the proton.

To maintain that 100 N force your foot must be in constant contact with the proton during that 1/100 second interval. Let's see what Newtonian mechanics says happens. Assume your foot has found a proton that is at rest with respect to your center of mass. In that 1/100 of a second interval where you push on the proton with a force of 100 N, your foot will have pushed that proton 300+ light years behind you.

So, there is one tiny problem here: Your leg needs to telescope to a length of 300+ light years in 1/100 second assuming Newtonian mechanics. There's another tiny problem here: Things can't travel 300+ light years in 1/100 second in our relativistic universe. Fortunately, relativity comes to our aid. That external force on the proton has less and less effect on the proton's speed as the proton's speed approaches the speed of light. Instead of needing your leg to be able to telescope to a length of 300+ light years in 1/100 of a second, you only need to have a leg that can telescope to a length of about 3,000 kilometers in that short time period.

BruceW
Homework Helper
Yes, if he redefines F*dt as an instantaneous impulse given to the proton, then it works out the same. So the only thing he's done 'wrong' is to assume that we have the technology to make such a thing, right?

Edit: (it doesn't necessarily need to be an instantaneous impulse, but an impulse which is over a very small amount of time, as if he had a hand-held particle accelerator.)

D H
Staff Emeritus
Yes, if he redefines F*dt as an instantaneous impulse given to the proton, then it works out the same. So the only thing he's done 'wrong' is to assume that we have the technology to make such a thing, right?

Edit: (it doesn't necessarily need to be an instantaneous impulse, but an impulse which is over a very small amount of time, as if he had a hand-held particle accelerator.)
That's no hand-held particle accelerator. It is a 3000+ kilometer long, 2000 yotta electron volt proton accelerator. And no, we don't have the technology to make such a device.

BruceW
Homework Helper
There isn't any theoretical lower limit on the length of the particle accelerator, but you're right that current technology would not be able to make a hand-held particle accelerator for those kinds of energies (which is one of many reason's why the 'running in space' example is not practically possible)

Thanks all. From what I gathered, my force and time of interaction are too horribly skewed for the situation. What I did instead was to calculate the force as dependent on the change in velocity of the proton. Saying it went from rest to a conservative (non-relativistic) 0.1c over the time step dt gives a force on the order of 10^-18 N per particle; much, much smaller than my original assumption of 100N.

Even if upgrading to solar wind densities and keeping the 0.1c speeds, treating the entire body as a wall to catch protons (ignoring radiative energy), it takes a large number of years to move a meter.

It also seems, from this link http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20030093608_2003101292.pdf, that the radiative energy is orders of magnitude higher and makes this already silly concept of running even more negligible. You're better off splaying your limbs and surfing your way out.

sophiecentaur