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# How do you use the Rocket Equation?

## Question

Hallo folks - this question is to help me improve my knowledge of not simply things Kerbal, but things educational. I know this is going to sound odd since as far as I can see the Rocket Equation is pretty basic math - the problem is that though I have many sterling qualities* math has never been one of them. I've always gravitated much more to technical knowledge, history, writing, etc.

(*Namely profound wisdom, devastating charm and Hollywood good looks. Oh - and fetching modesty.  )

The fact is, I'm mathematically inept - it's simply not something I ever really learned in my 50+ trips around the Sun. I look at this:

...and I can see, in principle, that it should be a relatively simple matter of arithmetic - which is the absolute limit of my mathematical knowledge. But applying it leaves me scratching my head. Where do these values come from? How do I work this?

What I would like to ask and I hope no-one minds: Would someone be willing to explain the Rocket Equation to me by taking me through using it, point-by-point?

Ideally, I'd love it if someone could build a simple rocket capable of a Munar landing and return, point out the correct values during each step so I can replicate the exercise in my own VAB - as a primarily visual and kinesthetic learner this method works much better for me than a simple description.

Sorry - I know that on the scale of learning this is about as basic as making ice cubes but it is something I'd love to learn a little better.

Cheers!

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Scott Manley did a great video on this:

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1 minute ago, GoatRider said:

Scott Manley did a great video on this:

Oh thanks! Lol I should have looked up Scott first thing; watching it now.

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52 minutes ago, NorthernDevo said:

Imagine you're floating in deep space with a ball, and you throw it as fast as you can. What happens? (Assume for the sake of argument that you throw it so gracefully that you don't end up spinning around.)

The momentum of a body is its mass times its velocity, and this is a conserved quantity. That means that your momentum plus the baseball's momentum is the same before and after your throw. Suppose we decide to measure everything in your original reference frame, so your initial velocity is 0, and therefore the total momentum is also 0. After the throw, your velocity and the ball's velocity change, but the total momentum remains 0.

Massball * velocityball + massyou * velocityyou = 0

(where 'm' means mass, 'v' means velocity, 'b' means ball, 'y' means you, '0' means before, and '1' means after.)

If we want to figure out how much you accelerated, we can use some simple arithmetic to rearrange the terms:

velocityyou = -Massball * velocityball / massyou = -velocityball * Massball / massyou

So, your velocity after the throw is the same as the ball's velocity times the ratio of the masses. (The negative sign just means you and the ball are moving in opposite directions.)

The case of a rocket is similar but different. Instead of throwing one object one time, a rocket emits a continuous stream of propellant. Total momentum is still conserved, but the continuous nature of the propellant complicates things: the rocket's mass is also decreasing continuously, and the propellant that stays in the rocket partakes in the acceleration until it finally gets expelled. All of this requires the use of everyone's favorite tool for handling continuous quantities, the integral calculus. But at the end of that long road lies the familiar rocket equation, which still has the "ball"'s velocity and a ratio of masses, just in a bit different form.

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Here's a stab at a very non-mathematical explanation:

There are three input variables in the Rocket Equation: specific impulse, wet mass, and dry mass.*  Specific impulse is how efficient your rocket is at converting fuel into useful thrust.  Wet mass is fuel. Dry mass is all mass in your ship other than fuel (crew modules, science instruments, electrical systems, etc.).

Specific impulse has a pretty intuitive, positive relationship with delta-v.  More efficient rockets expel their exhaust at faster velocities.  Per Newton's third law, every action has an equal and opposite reaction.  Thus, throwing a given object (say, a kilogram or rocket exhaust) pushes your ship forward faster than you would get by throwing the same object very slowly.

Wet mass has a positive effect on delta-v.  As mentioned above, under Newton's third law, every unit of fuel you fire out the back of the rocket is going to impart some acceleration on your rocket (the exact amount depends on the speed of the exhaust and the mass of the rocket at that point, which are covered by the rest of the rocket equation.

Dry mass has a negative effect on delta-v.  This is because, all other things being equal, heavier objects are more difficult to move than lighter ones.  (This is Newton's second law).

So what is wet mass "good" and dry mass "bad"?  Two related reasons.  First, dry mass never does anything useful to generate propulsion, whereas fuel does.  Second, as fuel gets burned up, the remainder of the ship becomes lighter.  As mentioned above, a lighter ship is easier to accelerate.  So the rocket's acceleration increases during the course of a stage.  This means that the last second of burn time (or the last kilogram of fuel, if you prefer) provides more acceleration than the first second or kilogram did.

The mathematical formula the Rocket Equation uses to calculate delta-v is just a numerical version of these concepts.  As @HebaruSan describes above, you can get to the rocket equation using calculus, based on that idea that the rocket gets easier to accelerate as it burns away mass.  But you don't really need the math to employ the core concepts: to maximize delta-v you want lots of fuel, not much else, and a very efficient engine.

*The formula you quoted uses starting and ending mass.  Starting mass is wet mass plus dry mass; the ending mass is just the dry mass.   But I think wet and dry are conceptually easier to understand.

On 1/3/2019 at 5:55 PM, NorthernDevo said:

Ideally, I'd love it if someone could build a simple rocket capable of a Munar landing and return, point out the correct values during each step so I can replicate the exercise in my own VAB - as a primarily visual and kinesthetic learner this method works much better for me than a simple description.

I'm not really sure what you mean by "correct values for each step."  As you may have seen from the subway-style delta-v maps, different maneuvers (such as transferring from Kerbin to Munar orbit) take certain quantities of delta-v.  But the Rocket Equation does not say anything in particular about how you need to go about this.  You could do it with high-dry-mass, low-wet-mass, high-ISP ship (like a nuclear cargo tug).  You do it with a low efficiency engine like the Puff as long as you have an adequate fuel supply.  You can do it with multiple stages per maneuver, or cover several maneuvers with a single stage.

And then obviously, there are other requirements to a successful craft besides pure delta-v.  You need some dry mass in order to do something useful with your rocket (carry passengers or cargo, perform science, etc).  You need to have enough thrust at certain points, e.g., so your ship is capable of taking off from the surface of a body.  Higher ISP engines are good in isolation, but they tend to be heavier (which can hurt your delta-v due to dry mass), or not work at sea level, etc.

So really, the "correct" value of delta-v is enough to complete the maneuvers you need to make with the ship you need to fly.  Plus a decent safety margin for contingencies.

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Posted (edited)

That's quite enough serious answers, I'm in a strange mood.
Here's how to use the rocket equation:-

Borrow a piece of paper (till-roll, napkin, whatever) and write on it
"dV = (Isp.g).ln(Mwet/Mdry)"
Then explain that that means the more efficient the engine (Isp.g) and the more fuel you use (Mwet/Mdry) the more the rocket can accelerate (dV) but, because of diminishing returns caused by the mass of the fuel (ln) twice as much fuel doesn't mean it can accelerate twice as much.
Then ask her to write her phone number beneath it.

The rest is left as an exercise for the student.

Edited by Pecan
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Posted (edited)

Hi folks - thanks so much for all your help; I do have a better understanding of what it is and how it works now; thanks so much.

On 1/4/2019 at 11:10 PM, Aegolius13 said:

I'm not really sure what you mean by "correct values for each step."  As you may have seen from the subway-style delta-v maps, different maneuvers (such as transferring from Kerbin to Munar orbit) take certain quantities of delta-v.  But the Rocket Equation does not say anything in particular about how you need to go about this.  You could do it with high-dry-mass, low-wet-mass, high-ISP ship (like a nuclear cargo tug).  You do it with a low efficiency engine like the Puff as long as you have an adequate fuel supply.  You can do it with multiple stages per maneuver, or cover several maneuvers with a single stage.

Now this, I fear, is something that happens a lot to me whenever I ask about math because - as seen in the received answers - the question I wrote was the wrong one; it was not what I was trying to learn. @Aegolius13 got to the heart of my confusion with this comment and this helps me to frame the question a little better. That said; all of the above is extremely valuable and I'm very happy to be learning this part of the equation.

In response to another question elsewhere, @Kryxal said:

Quote

Here's how I figure things to be:

• 3500 m/s to LKO
• 860 m/s transfer orbit
• 280 m/s Mun orbit
• 600 m/s plus LARGE margin for landing
• 600 m/s back to orbit
• 280 m/s transfer to suborbital trajectory

I was incorrectly thinking of the Rocket Equation for this - because I didn't know precisely what it did or how it is applied. My question actually is how does one obtain the numbers shown above when planning a flight? How do I know how much Delta-V I'll need to reach - say - Minmus without using my usual "Overbuild and Overpower" tendency? Aegolius said "As you may have seen from the subway-style delta-v maps, different maneuvers (such as transferring from Kerbin to Munar orbit) take certain quantities of delta-v." Well - I haven't actually seen them, but now that I know they're there, I'll certainly be looking.  I did think that by using the rocket equation one could figure that out.

Thanks so much for your help folks; this really is fascinating.

I'm trying a new career on 'hard' mode (No revert, no cash to start, no MJ for transfers, life support and a few other conditions in there) so the need to have a much better understanding this stuff is really making itself felt - I have to really plan and prep as carefully as possible and I'm enjoying every moment of it.

Cheers!

Edited by NorthernDevo

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Posted (edited)
1 hour ago, NorthernDevo said:

Well - I haven't actually seen them, but now that I know they're there, I'll certainly be looking.  I did think that by using the rocket equation one could figure that out.

The standard dV map is at: https://i.imgur.com/gBoLsSt.png

The rocket equation, as you now know, tells you how much dV a rocket can generate, so in mission-planning it will only tell you what it is capable of.
If you want to work out the values on the dV map yourself what you need to know is how to calculate orbital velocity for given altitudes and then the transfer cost.  Since you know how far Mun is from Kerbin, for instance, you can then find the difference in Kerbin low-orbit and high-orbit (at Mun altitude).  Alternatively, work through the calculations for Hohmann transfers, which will lead back to the orbital velocity calculations.

Edited by Pecan
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Posted (edited)
2 hours ago, NorthernDevo said:

My question actually is how does one obtain the numbers shown above when planning a flight? How do I know how much Delta-V I'll need to reach - say - Minmus without using my usual "Overbuild and Overpower" tendency?

"what you need to know is how to calculate orbital velocity for given altitudes and then the transfer cost." <-- mathy method that always underestimates what it really takes to do anything.

Or: you create an overbuilt and overpowered rocket, note the amount of dV you have to start with, go where you want to go, note how much dV you end with, and subtract. Then you call that a "computer simulation", write down the number, revert the flight, and do it again for reals.

Edited by bewing
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Posted (edited)
9 hours ago, NorthernDevo said:

My question actually is how does one obtain the numbers shown above when planning a flight? How do I know how much Delta-V I'll need to reach - say - Minmus without using my usual "Overbuild and Overpower" tendency?

The values in the delta-v charts can be derived from the "vis-viva" equation, which is also the basis of the "Hohmann transfer"  equations for orbit changes.

The vis-viva equation gives the current velocity at any point in an elliptical orbit, based on the current altitude.

I can't insert an image from this computer, but the equation is v = sqrt( GM * (2/r - 1/a)), where ...

• GM is the gravity constant for the body you're currently orbiting
• 'r' is the current altitude (plus the planet's radius)
• 'a' is the semi-major axis of the orbit (which can be computed as the average of the Ap and Pe values, plus the planet radius).

This works for all elliptical orbit changes.

The initial delta-v for getting into orbit is a given value of 3500 m/s, based on experience. The presence of the atmosphere prevents vis-viva from being used for the initial orbit.

So flying out from Kerbin you can work out the delta-v changes by knowing what orbit you're currently in (that will be the Pe), and what altitude you're aiming for (that will be the new Ap).

Example:

I'm in a circular low-Kerbin orbit at 80 km, so Ap = Pe = 80 km. Vis-viva says my speed is 2,279 m/s. (Note that the launch rule-of-thumb says I used 3,500 m/s to get there).

If I want to reach the edge of the Mun's SOI, that would be a new Ap of (Mun's orbit - Mun's SOI size) = (12M meters - 2.43M meters) = 9.57M meters.

Update the 'a' value using this new Ap value. Use the current Pe (80 kM) to compute the current altitude. Plug it into vis-viva, and the speed I need to be going right now to reach that Ap is 3,120 m/s. Since I'm currently going 2,279 m/s,  I need to burn prograde by the difference (about 840 m/s).

The delta-v charts typically say that a Mun intercept costs 860 m/s. So that's a good estimate.

When computing delta-v values, you need to use the GM value for that body. So when in the Mun SOI, you use a different GM than when orbiting Kerbin, etc.

Note that the vis-viva equation doesn't cover changes of orbit inclination.

Re: the rocket equation:

The rocket equation can be used to design a rocket that gives the delta-v you need to fly those trajectories.

Edited by FloppyRocket
Moar clean-up
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1 hour ago, FloppyRocket said:

The values in the delta-v charts can be derived from the "vis-viva" equation, which is also the basis of the "Hohmann transfer"  equations for orbit changes.

The vis-viva equation gives the current velocity at any point in an elliptical orbit, based on the current altitude.

I can't insert an image from this computer, but the equation is v = sqrt( GM * (2/r - 1/a)), where ...

• GM is the gravity constant for the body you're currently orbiting
• 'r' is the current altitude (plus the planet's radius)
• 'a' is the semi-major axis of the orbit (which can be computed as the average of the Ap and Pe values, plus the planet radius).

This works for all elliptical orbit changes.

The initial delta-v for getting into orbit is a given value of 3500 m/s, based on experience. The presence of the atmosphere prevents vis-viva from being used for the initial orbit.

So flying out from Kerbin you can work out the delta-v changes by knowing what orbit you're currently in (that will be the Pe), and what altitude you're aiming for (that will be the new Ap).

Example:

I'm in a circular low-Kerbin orbit at 80 km, so Ap = Pe = 80 km. Vis-viva says my speed is 2,279 m/s. (Note that the launch rule-of-thumb says I used 3,500 m/s to get there).

If I want to reach the edge of the Mun's SOI, that would be a new Ap of (Mun's orbit - Mun's SOI size) = (12M meters - 2.43M meters) = 9.57M meters.

Update the 'a' value using this new Ap value. Use the current Pe (80 kM) to compute the current altitude. Plug it into vis-viva, and the speed I need to be going right now to reach that Ap is 3,120 m/s. Since I'm currently going 2,279 m/s,  I need to burn prograde by the difference (about 840 m/s).

The delta-v charts typically say that a Mun intercept costs 860 m/s. So that's a good estimate.

When computing delta-v values, you need to use the GM value for that body. So when in the Mun SOI, you use a different GM than when orbiting Kerbin, etc.

Note that the vis-viva equation doesn't cover changes of orbit inclination.

Re: the rocket equation:

The rocket equation can be used to design a rocket that gives the delta-v you need to fly those trajectories.

Yikes! That's a lot of math!

Just kidding - thanks so much; your example helps a lot.

Looking at the equation initially my brain went 'wubba' but your explanation seems quite clear and easy to follow - I'll try doing this right away.

Ultimately, I'd like to be able to do the calculations just like this for - for example - a Duna transfer, though I suspect that it'll take a little time to get comfortable with using this.

I'm sorry - I'm sure this is pretty basic stuff mathematically speaking; in high school I was more interested in arts, history and the girl in the next seat.  I've always regretted not having a better understanding of math. No better time to start learning, I figure!

Cheers!

9 hours ago, Pecan said:

The standard dV map is at: https://i.imgur.com/gBoLsSt.png

The rocket equation, as you now know, tells you how much dV a rocket can generate, so in mission-planning it will only tell you what it is capable of.
If you want to work out the values on the dV map yourself what you need to know is how to calculate orbital velocity for given altitudes and then the transfer cost.  Since you know how far Mun is from Kerbin, for instance, you can then find the difference in Kerbin low-orbit and high-orbit (at Mun altitude).  Alternatively, work through the calculations for Hohmann transfers, which will lead back to the orbital velocity calculations.

Thanks so much - I'm having a great time learning this aspect of KSP. Very likely I'll use the subway map for most of my flights, though I would definitely like to take the opportunity to learn the mathematics from which they derive.

8 hours ago, bewing said:

Or: you create an overbuilt and overpowered rocket, note the amount of dV you have to start with, go where you want to go, note how much dV you end with, and subtract. Then you call that a "computer simulation", write down the number, revert the flight, and do it again for reals.

Yeah; that's my normal approach although other than a few memorized numbers the 'writing things down' bit tends to escape me.

What I've done though is created a parallel game in Sandbox and when I have a test rig that might be a bit iffy, call it 'computer simulation' and try it there first. The number of times Jeb has been flattened, drowned, blown-up or crispy-fried in the past week has dropped remarkably.

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2 hours ago, NorthernDevo said:

when I have a test rig that might be a bit iffy

Yeah, well -- that's actually the fun part in the end. Years from now, you will still be creating iffy test rigs and needing to test them in sandbox before you risk strapping Jeb into them.

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3 hours ago, NorthernDevo said:

Yikes! That's a lot of math!

Just kidding - thanks so much; your example helps a lot.

Looking at the equation initially my brain went 'wubba' but your explanation seems quite clear and easy to follow - I'll try doing this right away.

Ultimately, I'd like to be able to do the calculations just like this for - for example - a Duna transfer, though I suspect that it'll take a little time to get comfortable with using this.

You shouldn't really need to any tricky math to put delta-v to work.  For a lot of stuff, you can tell what you need just by using maneuver nodes.  For example, say you're in orbit around Kerbin and want to go to the Mun.  You can just plunk a maneuver node and increase your prograde vector until your orbit gets out to a Mun intercept (or even just the Mun's altitude).  It should end up being around ~860m/s,  with a little variation depending on your starting orbit, what intercept you get, etc.  That's essentially what delta-v is in this case - you much you need to change your ship's velocity to get from one orbit to the next.

Some situations can be a little trickier.  For example, when landing or taking off, you have to deal with gravity losses and (if the planet has an atmosphere) that will eat up some of your delta-v.  This is why orbital velocity around Kerbin is ~2,200 m/s, but it takes more like 3,000m/s of delta-v (if not more) to actually reach orbit.  The subway maps have reasonable estimations of these numbers, but actual results can vary a bit depending on your ship's thrust, aerodynamics, and your flight profile.  All are good reasons to carry a little spare fuel for safety margins.

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12 hours ago, NorthernDevo said:

Thanks so much - I'm having a great time learning this aspect of KSP. Very likely I'll use the subway map for most of my flights, though I would definitely like to take the opportunity to learn the mathematics from which they derive.

The map is there so you don't need to learn the maths (I never have, although I looked at it to make sure I understood it).  The rocket equation is fairly easy although, again, I use KER or MJ (mods) to work it out for me these days.  Find the dV you need from the map, build a rocket with at least that much from the rocket equation.  Make sure you have enough TWR to launch - everything else is down to how you fly it ^^.  Good luck.

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