# What is Rocket equation: Definition + 63 Threads

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. ### I Deriving the relativistic rocket equation with exhaust efficiency

Hi, I'm looking at this relativistic rocket equation on Wikipedia. Something doesn't make sense here, and I can't find a derivation for this equation in the linked source, so I'm trying to derive it myself with limited success...
2. ### How does burn time affect the parameters in the rocket equation?

The items (a) and (b) are provided for context. I am not concerned with solving the problem. That is relatively easy. My question is about the burn time. It doesn't seem to matter for solving the problem as it has been posed. All we care about is the states at the beginning an end of each...

11. ### How tyrannical is the rocket equation for a human RT to Mars?

I was reading this: https://www.cnbc.com/2020/12/01/elon-musk-highly-confident-spacex-will-land-humans-on-mars-by-2026.html It just seems that the Delta-V requirements - especially with a spacecraft that is stocked with supplies for a few years or for the Hohmann transfer helio-orbits - will...
12. ### Sign problem in the rocket equation

Question: Solution: Issue: I would like to know why ##\Delta m=-\Delta M## rather than ##=\Delta M## if minus signs have been used in the second equation. Thank you.
13. ### Rocket Problem -- Solve using the Rocket Equation

I tried the second rocket equation vf = vi + v rel * ln(Mi/Mf) but it gives out approximately 4900 m/s for the answer but the answer is 4160 m/s
14. ### How Does the Relativistic Rocket Equation Describe Velocity?

Show that, according to relativistic physics, the final velocity ##v## of a rocket accelerated by its rocket motor in empty space is given by ##\frac{M_i}{M} = \Big ( \frac{c+v}{c-v} \Big) ^ \frac{c}{2 v_{ex}}## where ##M_i## is the initial mass of the rocket at launch (including the fuel)...
15. ### How to model a rocket equation from the derivative of momentum?

I am using the derivative of momentum (dp/dt) with Newton’s 3rd Law with the gravitational force of Earth. F - [Force of gravity on rocket] = dp/dt F - (G * m_e * m_r / r2 ) = v * dm/dt + ma F = Force created by fuel (at time t) G = Gravitational Constant m_e = Mass of Earth m_r = Mass of...
16. ### Can Lagrangian Method be Applied to Solve Rocket Motion Equations?

Homework Statement While solving equation of rocket motion with Newton's law in 1-d,I pondered to apply Lagrangian method on this. However, I didn't get correct result. Because I can eliminate last 2nd equation using last equation and get some other equation which is certainly not rockets...
17. ### Error When Deriving the Rocket Equation

Homework Statement I want to try to derive the rocket equation and then add additional effects like gravity, air resistance, etc. Here's the equation that I found online: Homework Equations p (momentum) = mass * velocity F (force) = Δp / Δt The Attempt at a Solution [/B] pi = mv Here is...
18. ### Rocket equation without infinite small variables

Hey, I'm tutor for theoretical physics for first year students and I found a question that I couldn't answer so far. It's about the rocket equation. I tried to derive the acceleration without using infinite small variables, but somehow there is one term left that shouldn't be there. In the...
19. ### B Is the Tsiolkovsky Rocket Equation Worth Tattooing?

I recently thought about getting it as a tattoo, and thought I'd learn a little more since you know, I might be tattooing it on me lol and I really haven't done much research other than a couple first page Google results so far. I figured this would be a good place to start and learn some...
20. ### I How to develop the rocket equation?

I am referring specifically to chapter 3.2 in John Taylor's textbook Classical Mechanics where he develops the equation of motion for a rocket, but not in enough detail for me. Taylor first writes down the change in the momentum of the rocket as dP = P(t + dt) - P(t) = (m + dm)(v + dv) - mv =...
21. ### Calculating Fuel-Payload Ratio for Martian Rocket Lift-off

Watching "Mars" on National Geographics I try to calculate what the cost would be to bring anything usefull back to Earth. More specifically I wonder what fuel-payload ratio a rocket could have when lifting off from Mars. I want to use Tsiolkovsky's rocket equation Δv = ve ln(m0/mf) or Δv =...
22. ### I Multistage continuous Rocket Eqn

So if you have a rocket let's say that discards all the structural and engine mass continuously at zero velocity that is relative to the rocket until only the payload is traveling at the final velocity - then what will the equation of motion will look like? we can neglect the drag and...
23. ### Maximizing rocket velocity - shoot fuel at once, or slowly?

I stumbled upon a 3-year old article from Wired that poses this question on rockets: Suppose I have two rockets with a mass M and fuel mass m. Rocket A shoots all the fuel at once (again, like a nuclear propulsion engine) with a fuel speed of u and rocket B shoots two blobs of fuel—first a shot...
24. ### Relative velocity of exhaust in Ideal Rocket Equation

In this derivation of the Ideal Rocket Equation (https://en.wikipedia.org/wiki/Tsiolkovsky_rocket_equation#Most_popular_derivation), they use the fact that ##V_e = V - v_e##, or that the velocity of the exhaust in the observer frame ##V_e## is the velocity of the rocket ##V## minus the speed of...
25. ### A rocket needs to hover stationary over the ground

Homework Statement A rocket (initial mass ##m_o##, constant exhaust velocity ##v_{ex}## needs to use its engines to hover stationary, just above the ground. a) If it can afford to burn no more than a mass ##\lambda m_o## of its fuel (##\lambda \lt##), for how long can it hover? b) If ##v_{ex} =...
26. ### Deriving the relativistic rocket equation

Homework Statement so i read morin's derivation of rocket equation propelled by photons now i want to try for relativistic mass ejection but i am having some problems let subscript e denote quantities of ejected material and subscript of r denote quantities of rocket Homework Equations ## P =...
27. ### Rocket thrust force calculation

Homework Statement A rocket has weight 10,000 kg and it needs to carry 5000 kg with acceleration 3.5 m/s². How much thrust force is necessary for the rocket to have a flight from Earth to space? Homework Equations F=ma[/B]The Attempt at a Solution I try that, According to Newton's law, (...
28. C

### I Rocket equation: is there an optimal thrust?

This is my first post. I looked through many threads and could not find an answer to the question below. Well known equations in rocketry are: Delta Velocity = Ejection Velocity * ln (Final Mass / Initial Mass ) and Net Thrust = Mass Ejection Rate * Ejection Velocity – Current Mass *...
29. ### What is the Initial Mass of Fuel Needed for a Rocket to Reach 0.1c?

The problem states: Typical chemical fuels yield exhaust speeds of the order of 103 m/s. Let us imagine we had a fuel that gives v0 = 3 × 105 m/s. What initial mass of fuel would the rocket need in order to attain a final velocity of 0.1c for a final mass of 1 ton? I derived the equation in the...
30. ### Derivation of Rocket Equation Using Relative Velocity

Based on my current understanding of the problem I do not see this following derivation as valid, although this is what was given in my course notes. Although this particular example is from an undergraduate physics course this is not a homework problem: I'm confused about the underlying...
31. T

### Relativistic rocket equation (intuitive derivation)

Homework Statement In Newtonian mechanics, the rocket equation is derived by solving the simple differential equation -dm U = m dV, where U is the velocity of the expelled material relative to the rocket; a matter of conservation of momentum. In order to get the correct relativistic equation...
32. ### B Separation of variables - rocket equation

hello there Im trying to do a derivation of tsiolkovsky's rocket equation, but i got stuck at the step when i have to use separation of variables (marked with red in the pic), i used maple to solve it, so i could get on with it, but i want to understand what is happening to solve this, so can...
33. ### Find velocity of rocket that obtains more gas from gas cloud

Homework Statement To solve this problem you will need to construct a differential equation. A picture of the situation will help. Ignore all gravitational forces. A Rocket ship of structural mass M and fuel mass m, begins at rest relative to a gas cloud. The ship burns fuel at the rate ω which...
34. ### I Why are some systems defined with the name "variable mass"?

I do not get why systems such as the rocket in space are defined as "variable mass" since the mass of the system is not varying. The equation used for such systems $$\sum F^{(E)}=\frac{d\vec{P}}{dt} \tag{1}$$ (sum of external forces on the system equals the change in momentum) holds true only...
35. ### How long does it take for a rocket to lift off the ground?

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...
36. ### Calculating Kinetic Energy and Maximum Speed for a Leaking Oil Tanker Truck

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 =...
37. ### How Does a Relativistic Photon Rocket Calculate Acceleration and Speed?

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 EquationsThe Attempt at a Solution Part(a) 4-vector acceleration is given by ##\gamma^2 \left[...
38. ### What is the derivation of 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...
39. ### 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) +...
40. ### 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...
41. ### 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...
42. ### What is the mistake in the derivation of the 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...
43. ### 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?
44. ### 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...
45. ### Understanding the Relativistic Velocity Addition Formula

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...
46. ### 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...
47. ### Rocket Equation Homework: Explaining Mass Infinity

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 changee^1 = 2.72 e^2 = 2.74 e^3 = 20.4 As ΔV/C goes up, the mass of the spacecraft goes up faster...
48. ### Where can I find the formula for the 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...
49. ### What Is the Velocity of a Leaking Bucket at the Moment It Empties?

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...
50. ### 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...