# Rocket equations of motion w/ drag and gravity

In summary, John Taylor's two examples of linear drag and gravity can be combined to give a formula for the velocity of the rocket before burnout. This works for the case of gravity but no drag.
I have seen many examples of the EOM for a rocket derived for the following cases:
• No gravity, No drag
• Gravity, No drag
• No gravity, linear drag (b*v where b is a constant)

I have never seen
• Gravity, linear drag

I took John Taylor's two examples of linear drag and gravity (without drag), and tried to combine them (since quadratic drag would add even more complexity). A couple basic assumptions at first:
The mass of the rocket is described as:
$$m(t)=m_o-kt$$
Where $m_o$ is the initial mass of the fueled up rocket, and $-k$ is the rate at which mass leaves the rocket. Also, $u$ is the velocity of the exhaust (which is taken as a constant), and $v$ is the velocity of the rocket itself.

For linear drag + gravity, the resultant equation of motion would be:

$$\frac{m dv}{dt}=-\frac{u dm}{dt}-mg-bv$$

At this point, for the case without gravity, you can do separation of variables. However, that doesn't seem possible here:
$$\frac{dv}{dt}=\frac{1}{m}\left(ku-bv\right)-g$$
(I've substituted -k in for dm/dt)
So, putting in m->m(t):
$$\frac{dv}{dt}=\frac{\left(ku-bv\right)}{\left(m_o-kt\right)}-g$$

Any help on solving this analytically would be greatly appreciated!

Just doing this in my head, so it might not work...
Looks like you can get it into the form y'=Ay/x+B. Then try substituting y=ux.

I finally found a method in Boyce & DiPrima's diff eq book:
If you can get the diff eq to look like:
$$v'(t)+p(t)v(t)=g(t)$$
Then you can solve it using an integrating factor, $\mu (t)$:
$$\mu(t)=exp\int p(t)dt$$
Then,
$$v(t)=\frac{1}{\mu(t)}\int_{t_o}^{t}\mu(s)g(s)ds$$
In my case, $t_o=0$ which helped simplify it. Anyway, I solved and got a somewhat ugly but completely reasonable formula!
$$v(t)=\frac{ku}{b}\left[1-m_o^{-b/k}\left(m_o-kt\right)^{b/k}\right]-\frac{g}{b-k}\left[m_o-kt-m_o^{1-b/k}\left(m_o-kt\right)^{b/k}\right]$$

Which gives the velocity of the rocket anytime before burnout. I can integrate and get height, and then use conservation of energy (with quadratic drag this time) to calculate final height. Sweet.

Extra cool:

I used mathematica to take the limit of that equation as $b\rightarrow0$ and got exactly the same result as John Taylor's 3.11 which solves for $v(t)$ for the case of gravity but no drag.

Mind = blown. So happy this worked.

(also, same for $g\rightarrow0$, recovers the same as 3.14)

Last edited:
berkeman
The method I outlined works too. Did you try it?
Writing T=(m0/k)-t and V=(ku/b)-v we have dv=-dV, dt=-dT:
dV/dT=(b/k)(V/T)-g
Now putting V=WT
W' T + W = (b/k)W - g
Writing α=b/k-1
W' T = αW - g
W'/(αW-g)=1/T
Etc.

I did not try that method, since I couldn't get it into the form you proposed:
$$v'=\frac{Av}{t}+B$$

The best I can do with that initial equation is:
$$v'=v\left(\frac{-b}{m(t)}\right)+\frac{uk}{m(t)}-g$$
or
$$v'=-vp(t)+g(t)$$

Does your method work for that form as well?

I couldn't get it into the form you proposed:
My post #5 lays it out in detail.

haruspex said:
My post #5 lays it out in detail.
Wow that is really clever. At first I didn't get what you were saying but yeah that works! I would have never thought of that on my own...

## 1. What are the basic equations for calculating the motion of a rocket with drag and gravity?

The basic equations for calculating the motion of a rocket with drag and gravity are Newton's Second Law of Motion, the Drag Equation, and the Gravitational Force Equation. These equations can be used to determine the acceleration, velocity, and position of a rocket as it travels through the atmosphere.

## 2. How does drag affect the motion of a rocket?

Drag is a force that opposes the motion of an object through a fluid (such as air). In the case of a rocket, drag is caused by the air resistance as the rocket moves through the atmosphere. This force can slow down the rocket's acceleration and decrease its velocity.

## 3. What is the role of gravity in the equations of motion for a rocket?

Gravity is a force that pulls all objects towards the center of the Earth. In the equations of motion for a rocket, gravity is a key factor in determining the rocket's trajectory and acceleration. As the rocket moves away from the Earth's surface, the force of gravity decreases, allowing the rocket to travel further.

## 4. How do the rocket equations of motion change at different altitudes?

The equations of motion for a rocket with drag and gravity will change at different altitudes due to changes in air density and gravitational pull. At higher altitudes, the air density is lower, resulting in less drag force. Additionally, the force of gravity decreases as the rocket moves further away from the Earth's surface.

## 5. What are some real-world applications of the rocket equations of motion with drag and gravity?

The rocket equations of motion with drag and gravity are used in the design and testing of rockets and spacecraft. They are essential for determining the trajectory, stability, and overall performance of a rocket during its launch and flight. They are also used in atmospheric and spaceflight simulations to predict and analyze the behavior of rockets and spacecraft in various environments.

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