How Is Acceleration Calculated for Space Travel Via Cannon?

[aiop]

The distance from the Earth's surface to the boundary of space is 100km. What i was wondering is how to calculate the acceleration you would need, incorporating air resistance and gravity to travel this distance. If you were being launched by a canon.

So the initial acceleration, if someone knows the formulas that you could use that would be awesome, thanks! :)

 

[PeroK]

The distance from the Earth's surface to the boundary of space is 100km. What i was wondering is how to calculate the acceleration you would need, incorporating air resistance and gravity to travel this distance. If you were being launched by a canon.

So the initial acceleration, if someone knows the formulas that you could use that would be awesome, thanks! :)

Try googling for "escape velocity".

 

[BvU]

Google escape velocity

[edit] Ha !

Note that it's not a matter of acceleration: a simple $g$ is enough to get started

 

[russ_watters]

If you are using a cannon, the acceleration is a function of the length of the barrel.

 

[jbriggs444]

If you are using a cannon, the acceleration is a function of the length of the barrel.

If the cannon is 100 km in height, one can call it an "elevator" and use negligible acceleration.

 

[Bystander]

https://en.wikipedia.org/wiki/Gerald_Bull
... or, more "realistically" ...

 

[litup]

The distance from the Earth's surface to the boundary of space is 100km. What i was wondering is how to calculate the acceleration you would need, incorporating air resistance and gravity to travel this distance. If you were being launched by a canon.

So the initial acceleration, if someone knows the formulas that you could use that would be awesome, thanks! :)

If you are actually thinking of sending humans up that way, forget it. Suppose you had a 100 meter long cannon. Escape velocity is about 40,000 km/hr or about 11 km/second. It would take at least twice that velocity out the cannon to get to escape velocity because of atmospheric friction. Let's suppose you could achieve escape by doubling the cannon velocity to 22 km/second you would need to get to in 100 meters. The formula says it would take about 2000 g and the time in the cannon would be about 1/10th of a second. Of course, if a human was inside, she would be a mass of bloody skin and bones at the bottom of the vehicle and be quite dead before leaving the cannon.

But if you did 2000 g's for 1/10th of a second you would be up to about 80,000 km/hr in 100 yards. That would be double escape velocity and you would also have to have extremely good insulation to keep from burning up the craft before it ever reached 100 km.

Not many electronic boxes can take 2000 g's either. You could imagine sending fruit up that way, it would puree itself all over the bottom of the craft.

You could certainly send metal that way or other materials but not anything you want to be alive when it leaves the cannon.

Which is why they use very large rockets which limits acceleration to about 3 g's so humans can survive entry into orbit.

 

[aiop]

If you are actually thinking of sending humans up that way, forget it. Suppose you had a 100 meter long cannon. Escape velocity is about 40,000 km/hr or about 11 km/second. It would take at least twice that velocity out the cannon to get to escape velocity because of atmospheric friction. Let's suppose you could achieve escape by doubling the cannon velocity to 22 km/second you would need to get to in 100 meters. The formula says it would take about 2000 g and the time in the cannon would be about 1/10th of a second. Of course, if a human was inside, she would be a mass of bloody skin and bones at the bottom of the vehicle and be quite dead before leaving the cannon.

I was wondering how to do the math because you would half to be in a constant state of acceleration. Wondering how to do that math taking gravity and air resistance into account.

Try googling for "escape velocity".

Also isn't escape velocity the minimum velocity you would need to leave Earth implying a constant velocity. I am taking about the velocity it would take with one single push that it would take to leave earth.Thanks every one for the replies !

 

[jbriggs444]

I was wondering how to do the math because you would half to be in a constant state of acceleration. Wondering how to do that math taking gravity and air resistance into account.

1. Set up a differential equation which includes air resistance as a function of altitude and velocity and gravity as a function of altitude.
2. Solve said equation.

The air resistance will depend on the size and shape of your projectile. Gravity will depend on its mass.

It is likely to be a difficult equation to solve. So a numerical approach may be easier. Solve the problem in time-reversed fashion. Drop the projectile at zero velocity at the edge of the atmosphere and iterate backwards in time as air resistance and gravity both accelerate it toward the Earth. Read out the final velocity just prior to impact.

 

[BvU]

I was wondering how to do the math because you would have to be in a constant state of acceleration. Wondering how to do that math taking gravity and air resistance into account.

You would not necessarily have to be in a constant state of acceleration (at least: not upwards ..). Witness the term escape velocity. But, just for assistance purposes, are you familiar with the basic equations for projectile trajectories ? SUVAT and such ?

 

[Tom.G]

Drop the projectile at zero velocity at the edge of the atmosphere and iterate backwards in time as air resistance and gravity both accelerate it toward the Earth. Read out the final velocity just prior to impact.

But that will give you the terminal velocity of free falling object, quite a bit lower than escape velocity (by a factor of approx. 25)

from: https://www.grc.nasa.gov/www/k-12/airplane/termv.html

 

[jbriggs444]

But that will give you the terminal velocity of free falling object, quite a bit lower than escape velocity (by a factor of approx. 25)

from: https://www.grc.nasa.gov/www/k-12/airplane/termv.html

We are not asked for escape velocity, but only the velocity to reach "the boundary of space". Escape velocity is irrelevant to the question at hand.

 

[BvU]

We are not asked for escape velocity, but only the velocity to reach "the boundary of space". Escape velocity is irrelevant to the question at hand.

Good point. Actually, post #1 asks for an acceleration -- which we transmogrified to velocity. Once poster understands escape velocity, the step to velocity to reach 100 km is a small step for man ...

 

[jbriggs444]

Good point. Actually, post #1 asks for an acceleration -- which we transmogrified to velocity.

Right. Calculating required acceleration in the cannon barrel is easy if you know the muzzle velocity (and barrel length, as @russ_watters has pointed out).

 

[aiop]

Good point. Actually, post #1 asks for an acceleration -- which we transmogrified to velocity. Once poster understands escape velocity, the step to velocity to reach 100 km is a small step for man ...

I was asking the question for the math point of view i know that its not escape velocity. I was asking with in 100km for an example, because i wanted to account for air resistance. I don't know the math of how to determine the initial velocity well accounting for an air resistance. That will be reduced as you go higher. Same with the variable of gravity.

SUVAT and such ?

Yes.. don't see how i could use those. In grade 12 so my knowledge is limited.

Thanks every one !

 

[jbriggs444]

[in response to a suggestion to use the SUVAT equations]
Yes.. don't see how i could use those. In grade 12 so my knowledge is limited.

Do you feel that you know enough to attempt a calculation assuming a fixed acceleration of gravity and negligible air resistance?

 

[aiop]

Do you feel that you know enough to attempt a calculation assuming a fixed acceleration of gravity and negligible air resistance?

Yes.

 

[BvU]

Same with the variable of gravity

Come to think of it, gravity at 100 km isn't much less than at 0 km height (100 km is only 1/60 th of Earth radius). So if you keep g constant things will simplify considerably. Ignoring air resistance, the SUVAT equations (or an energy balance) give you a value (1.4 km/s) to start with.

Then, using trial and error you could take air drag into account (my estimate: a hefty increase in velocity required) to work towards a ground-level muzzle velocity.

Biggest (range) guns built had 140 km horizontal range, so the odds you can build something that shoots upwards to 100 km aren't favorable. Methinks.