# Rocket Propulsion with gravity

• Lola Luck
In summary, The equation for the acceleration of a rocket is (dv/dt)=-vex/m(dm/dt), where mg is the force of gravity.
Lola Luck

## Homework Statement

Earlier we considered a rocket fired in outer space where there is no air resistance and where gravity is negligible. Suppose instead that the rocket is accelerating vertically upward from rest on the Earth's surface. Continue to ignore air resistance and consider only that part of the motion where the altitude of the rocket is small so that g may be assumed to be constant.

a) How is eq. (8.37) modified in the presence of the gravity force?

b) Derive an expression for the acceleration a of the rocket, analogous to Eq. (8.39)

## Homework Equations

eq (8.37): m(dv/dt) = - vex(dm/dt)
m= mass of rocket vex= velocity of rocket exhaust v= velocity of rocket

eq (8.39): a= (dv/dt) = (-vex/m)(dm/dt)

## The Attempt at a Solution

For part (a) I believe I need to factor in the impulse of gravity, Jg= mg(dt), but I don't know how. Then I would use my answer to part (a) to find an equation for acceleration.

Thank you

Lola Luck said:
For part (a) I believe I need to factor in the impulse of gravity, Jg= mg(dt), but I don't know how.
You need to include the effects of gravity, but not as you've stated. Try looking at units for 8.37 and rethink what you "believe."

i believe that the trick lies in realising the fact that the net external force acting on the rocket is now, no longer the reaction due to its own engines, but also gravity acts on it. so we need to modify the LHS first

sorry for the late response!

so the units of the equation are kg*(m/s^2), which is the same as a Newton. So do I just add the force of gravity?

m (dv/dt)+mg=-v(dm/dt)

That'd be my first approach.

okay, and then I just solve for dv/dt to get acceleration for part (b). thank you!

## What is rocket propulsion with gravity?

Rocket propulsion with gravity is a method of launching a rocket into space by utilizing the Earth's gravitational pull. It involves using the Earth's gravity to slingshot a spacecraft towards its intended destination.

## How does rocket propulsion with gravity work?

Rocket propulsion with gravity works by taking advantage of the Earth's gravitational pull. A spacecraft is launched at a specific angle and speed in relation to the Earth's rotation, allowing it to gain momentum and accelerate as it moves towards its destination.

## What are the benefits of using rocket propulsion with gravity?

The main benefit of using rocket propulsion with gravity is that it requires less fuel compared to traditional rocket launches. This is because it utilizes the Earth's gravity to provide additional thrust, which reduces the amount of fuel needed for the launch.

## What are the challenges of using rocket propulsion with gravity?

One of the main challenges of using rocket propulsion with gravity is the precise calculations and timing required for a successful launch. The angle and speed of the spacecraft must be carefully calculated to ensure it uses the Earth's gravitational pull effectively, and any miscalculations could result in the spacecraft missing its target or losing momentum.

## Is rocket propulsion with gravity used in all space launches?

No, rocket propulsion with gravity is not used in all space launches. It is mainly used for interplanetary missions, where the spacecraft needs to travel long distances and conserve fuel. For shorter missions, traditional rocket launches are more efficient.

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