# Effect of Gravity on a Rocket Projectile

by Ron Kurtus (updated 29 May 2023)

The effect of gravity on a rocket projectile as it moves upward away from the Earth is that the force of gravity resists the thrust of the rocket engine, slowing the acceleration and resulting in more work having to be done.

Typically, the rocket is accelerated upward from the ground by the force of the rocket engine. This thrust must overcome resistances. A rocket can also be propelled upward at such a high velocity that it escapes the pull from gravity.

If the rocket is sent at an angle to the ground, it can go into orbit around the Earth. This situation will not be considered here.

If the engine is turned off during the flight, the rocket will start to slow down until it reaches a maximum altitude and then falls back to the ground.

The work done by the engine is a function of its thrust and the distance traveled.

Questions you may have include:

• What are the factors in rocket propulsion?
• What happens when the engine is turned off?
• What is the work done by the rocket engine?

This lesson will answer those questions. Useful tool: Units Conversion

## Rocket propulsion

A rocket consists of a rocket engine and a payload. When the fuel in the engine is ignited, chemical energy is converted into thermal energy and the expansion of gases. This creates the kinetic energy that produces the thrust of the rocket engine that propels the rocket upwards.

### Thrust must overcome resistances

The amount of thrust or force to propel the rocket upwards must overcome the resistance from gravity (weight of the rocket) and resistance from the inertia of the mass.

F = mg + ma

where

• F is the force propelling the rocket upwards
• m is the mass of the rocket (body and engine)
• mg is the resistance from gravity (weight)
• ma is the resistance from inertia to accelerate the rocket

Note that the mass of the rocket decreases with the consumption of fuel. For the sake of simplicity, we will not consider that in the equations.

### Acceleration

While the engine provides its thrust, the rocket will accelerate according to:

a = F/m − g

### Escape velocity

It is possible for the rocket to reach a high-enough velocity that it will escape from the pull of gravity and fly off into space. The escape velocity equation is:

ve = − √(2g[RE + h])

where

• ve is the required escape velocity in kilometers/second (km/s)
• RE is the radius of the Earth
• h is the height above the Earth's surface

Note: The negative sign in the equation indicates that the velocity vector is in a direction opposite of the gravitational force vector, according to our direction convention. Be aware that many textbooks state the equation as a positive value.

See Escape Velocity from Gravity

## Engine turns off

At some point in the flight, the engine may be turned off or run out of fuel. The rocket then starts to decelerate or slow down due to the pull of gravity. It finally reaches its maximum altitude and then starts to fall back to Earth.

### Maximum displacement

The equation for the peak or maximum displacement is:

ym = −vi2/2g

where

• ym is the maximum vertical displacement in m or ft
• vi is the initial velocity once the rocket engine is turned off
• g is the acceleration due to gravity (9.8 m/s2 or 32 ft/s2)

Note: the negative sign indicates the direction is away from the Earth

See Gravity Displacement Equations for Objects Projected Upward

### Energy at the peak height

At the peak height or displacement ym, the kinetic energy of the rocket KE = 0, since it is no longer moving upwards. However, its potential energy (PE) is a function of its height from the ground:

PE = mgym

(This equation is also often written as PE = mgh, where h is the height from the ground)

When the rocket starts to fall, its kinetic energy increases as the rocket speeds up.

## Work done by rocket engine

The work done by the rocket engine is the force or thrust times the upward distance or displacement at some given point or time in its flight. It can also be stated as the change in kinetic energy or change in potential energy.

### Work and displacement

The work done by the engine on the way up is:

W = Fd

where

• W is the work done by the rocket engine
• F is the force or thrust of the engine
• d is the displacement of the rocket at a given point in the flight

Since F = mg + ma,

W = md(g + a)

### Work with respect to change in energy

The work done can also be stated as the change in the rocket's kinetic and potential energy.

W = ΔPE + ΔKE

where

• ΔPE means change in potential energy
• ΔKE is the change in kinetic energy in J or kg-m2/s2

Since on the ground, PE = 0 and KE = 0, the work at some given displacement is:

W = mgh + mv2/2

## Summary

The effect of gravity on a rocket projectile is that it provides resistance to the upward motion of the rocket, as well as providing a downward accelerated force.

Once the engine is turned off during the flight, the rocket will start to slow down until it reaches a maximum altitude and then falls back to the ground.

The work done by the engine is a function of its thrust and the distance traveled.

## Resources and references

Ron Kurtus' Credentials

### Websites

Gravity Resources

Rocket Principles - NASA

Rockets - Physics - Wikipedia

Introduction to Rocket Propulsion - LumenLearning.com

Rocket Physics - Brilliant.org

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