# What is Rocket equation: Definition and 60 Discussions

The Tsiolkovsky rocket equation, classical rocket equation, or ideal rocket equation is a mathematical equation that describes the motion of vehicles that follow the basic principle of a rocket: a device that can apply acceleration to itself using thrust by expelling part of its mass with high velocity can thereby move due to the conservation of momentum.

Δ
v
=

v

e

ln

m

0

m

f

=

I

sp

g

0

ln

m

0

m

f

{\displaystyle \Delta v=v_{\text{e}}\ln {\frac {m_{0}}{m_{f}}}=I_{\text{sp}}g_{0}\ln {\frac {m_{0}}{m_{f}}}}
where:

Δ
v

{\displaystyle \Delta v\ }
is delta-v – the maximum change of velocity of the vehicle (with no external forces acting).

m

0

{\displaystyle m_{0}}
is the initial total mass, including propellant, a.k.a. wet mass.

m

f

{\displaystyle m_{f}}
is the final total mass without propellant, a.k.a. dry mass.

v

e

=

I

sp

g

0

{\displaystyle v_{\text{e}}=I_{\text{sp}}g_{0}}
is the effective exhaust velocity, where:

I

sp

{\displaystyle I_{\text{sp}}}
is the specific impulse in dimension of time.

g

0

{\displaystyle g_{0}}
is standard gravity.

ln

{\displaystyle \ln }
is the natural logarithm function.

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1. ### What's wrong with this math: A lot of thrust and almost no fuel?

So according to this if you need a force that can lift a human and even an extremely ultra-light craft (1500 N) and have a velocity of 200 m/s you can eject extremely small amount of fuel through the nozzle: 0.0007 or less kg per second for as long as you're using 15 MPa pressure which is doable...
2. ### Is this rocket equation right?

I tried to use conservation of momentum I only want to know if the equation I came up with is correct and if not, why?
3. ### I Is the classical ideal rocket thrust equation correct?

Hello! I have recently found this fascinating article: https://zenodo.org/record/3596173#.YJ1ttV0o99B The author claims that classical equation for rocket thrust in incorrect because F is not equal to ma for a changing mass. Neither my professors nor me can see any errors. Do you think this...
4. ### Exploring Electron Rockets for Interstellar Travel

Hi all, Why not build an electron rocket? Why won't this idea work? Seems like someone could just hitch an electron gun onto a spaceship. Boom! Electron Rocket. I was thinking about ion thrusters. These use ions to achieve thrust. They are problematic for achieving a high delta V because...
5. ### B Relativistic rocket equation

Hi guys. I am studying the relativistic implementation of the standard Tsiolkosvky rocket equation, but ran into some doubts. Doesn't special relativity apply only to inertial frames of reference(non-accelerated)? Then, how could it be used for the rocket equation, which speaks of a ##\Delta...
6. ### B Relativistic rocket equation

I am studying through online resources some principles of spacecraft propulsion, since it really fascinates me, and makes me want to know a bit more about it :) For rockets, thruster, I found the Tsiolkovsky rocket equation: $$\Delta v=v_e ln(\frac{m_0}{m_f})$$ Of course, rockets can travel...

32. ### Variable Mass of Rockets

Homework Statement A rocket with an initial mass of 60,000kg ignites its engines and burns fuel at a rate of 300 kg/s with an exhaust velocity of 2220 m/s. How long after the engines start does the rocket lift off the ground? Homework Equations From Newton's second law F = Ma this equation can...
33. ### Rocket Equation-esque Problem

Homework Statement A truck hauling a big tank of oil starts at position x=D (Chicago) and heads due west (–x direction) toward its destination at x = 0 (Des Moines). At Chicago, the total mass of the loaded truck is M and the mass of oil it is carrying is λM. (Thus M(1–λ) is the "tare" mass =...
34. ### Relativistic Photon Rocket

Homework Statement (a) Show ##a = \frac{a_0}{\gamma^3}##. (b) Find proper acceleration of rocket (c) Find speed as a function of time. (d) Find acceleration of second rocket. Homework Equations The Attempt at a Solution Part(a) 4-vector acceleration is given by ##\gamma^2 \left[...
35. ### Deriving the Rocket Equation

Hello, A little something I've been working on for fun today was trying to derive the rocket equation: Δv=V_{e}ln(R) So first I start with F=ma =m\frac{dv}{dt} \frac{F}{m}dt = dv ∫_{0}^{Tco} \frac{F}{m}dt = ∫_{0}^{v} dv Where Tco is the time of engine cutoff, and v is the velocity at that...
36. ### Variable mass calculation of thrust for a water rocket

Homework Statement I am trying to derive the formula for time varying thrust given that I know the initial mass/volume of water in a water rocket. Knowns will include initial pressure, initial volume of water, and nozzle cross sectional area. Homework Equations Bernoullis equation: (p/ρg) +...
37. ### Need help solving a differential equation numerically

Hello all, I am currently trying to solve a differential equation numerically. The equation is as follows: dv/dt = (u*q)/(m0-q*t) - g - (cd*ρ*A*0.5)*v2 If you haven't already guessed, it's the rocket equation with added gravity and drag. Now, I'm not even sure if that's what it's supposed to...
38. ### Simple Rocket Equation in Gravity

Homework Statement I'm having trouble deriving the equation for the velocity of a rocket in constant G given that it has constant exhaust velocity Homework Equations I know that a=dv/dt=U/M(dM/dt)-g The Attempt at a Solution so from here dv=U/M dM-gdt v=U ∫1/M dM - g ∫dt v from 0...
39. ### Tsiolkovsky rocket equation

Hello, I am trying to understand the Tsiolkovsky rocket equation. I am looking at this right now: http://www.math24.net/rocket-motion.html they said that the momentum of the rocket itself is: p1 = (m-dm)(v+dv) and the momentum of the gas was: p2 = dm(v-u) Here is the problem- u is...
40. ### How much is the ideal rocket equation affected by air drag?

http://en.wikipedia.org/wiki/Tsiolkovsky_rocket_equation ideal rocket equation. No fudge for atmospheric drag. how much is it a factor?
41. ### Rocket Equation and Orbit questions?

Homework Statement #1.) A rocket exhausts fuel with a velocity of 1500 m/s, relative to the rocket. It starts from outer space with fuel comprising 80 per cent of the total mass. When all the fuel has been exhausted the speed is... (Answer was 2400 m/s) Homework Equations Vrel...
42. ### Relativistic rocket equation

Homework Statement In this thread the author performs the following calculation under "method 1": v+dv=\frac{v+dv'}{1+vdv'}=v+(1-v^2)dv'\implies dv'=\frac{dv}{1-v^2} He's set c=1 so the second expression is the relativistic velocity addition formula. What I don't understand is how he gets the...
43. ### Tsiolkovsky's rocket equation question

How would the equation look if instead of knowing the effective exhaust velocity we knew the force the exhaust was exerting on the rocket. The equation is: \Delta V = v_e * ln(\dfrac{m_0}{m_1}) would \Delta V still be proportional to the log of the initial mass over the final mass...
44. ### Rocket equation

Homework Statement The following equation is known as the "Rocket Equation": \frac{M+P}{M}= e^{ΔV/C} = mass ratio M = dry mass P = mass of propellant C = exhaust velocity ΔV = velocity change e^1 = 2.72 e^2 = 2.74 e^3 = 20.4 As ΔV/C goes up, the mass of the spacecraft goes up...
45. ### Relavistic Rocket Equation

Hi all, Does anyone know where I can find the formula for the relavistic rocket equation? That is, I want to know the mass of fuel required to accelerate a rocket from 0 to β, for a given acceleration A, assuming the rocket (with no fuel) has a mass of M. All of this would be in the frame of...
46. ### Rocket equation: leaking bucket

Homework Statement A bucket of mass M (when empty) initial at rest and containing a mass of water is being pulled up a well by a rope exerting a steady force P. The water is leaking out of the bucket at a steady rate such that the bucket is empty after a time T. Find the velocity of the bucket...
47. ### Integration of Tsiolkovsky rocket equation

In the Tsiolkovsky rocket equation derivation there is a part that says: \frac{dV}{dt} = -\upsilon_e \frac{1}{m} \frac{dm}{dt} "Assuming v_e, is constant, this may be integrated to yield:" \Delta V\ = v_e \ln \frac {m_0} {m_1} How does this work? The differential is an operator...
48. ### Rocket equation exhaust velocity question

The rocket equation given by Delta V = Ve ln (Mo/M) says that Ve is the exhaust velocity of the rocket. The orbital velocity for Low Earth Orbit is about 7.8 Km/s. How then, is this velocity attained by a rocket which is fuelled only by a bipropellant rocket engine producing an exhaust...
49. ### Double stage Rocket Equation help

Homework Statement okay the problem is there is a two stage rocket with a payload of 200kg and isp 2943m/s. It has a total delta v of 10km/s and at each stage it produces half of the total delta v. The question is to calculate the initial mass prior to launch, I've been told also that for...
50. ### Rocket Equation with varying gravity

So, the rocket equation is F_ext = m(dv/dt) + u(dm/dt) where m is the mass of the rocket, v the velocity, u the effective exhaust gases speed, and F_ext the external forces on the system. If we take a constant mass ejection rate p, and take the external force to be the gravitational...