Rocket propulsion fuel velocity

[jack1234]

Rocket propulsion

In the serway textbook at the rocket propulsion section it is mentioned that
If the fuel is ejected with a speed v_e relative to the rocket (the subscript e stands for exhaust, and v_e is usually called the exhaust speed), the velocity of the fuel relative to the Earth is v-v_e.

May I know why the velocity of the fuel relative to the Earth is v-v_e?

 

[Astronuc]

Velocity is a vector and the velocity (v) of the rocket points away from Earth and is measured with respect to earth. v_e is measured with respect to the rocket and is in the opposite direction of travel. The velocity of the exhaust has a forward velocity v to which one adds -v_e to get the velocity with respect to earth.

It's like a person on a train, with the velocity v, who throws an object at velocity v_o. If v_o is forward with the train, the combined velocity (as would be observed by someone standing on a platform) is v + v_o. If the v_o is oriented to toward the rear (opposite v), then the combined velocity is v - v_o.

This also of a person in a boat traveling with v with respect to water, or a plane in the air with an air speed v. The effective velocity with respect to land (shore line or ground) is v +/- v_fluid, where v_fluid is the stream or air velocity. Here I use a simple example of parallel motion. It's a little more complicated -

http://hyperphysics.phy-astr.gsu.edu/hbase/boatc.html

http://hyperphysics.phy-astr.gsu.edu/hbase/airpw.html

http://hyperphysics.phy-astr.gsu.edu/hbase/relmot.html

 

[ogb p]

The velocity of the fuel relative to the Earth is v-v_e because of the principle of conservation of momentum. In rocket propulsion, the fuel is ejected at a high speed relative to the rocket, creating a thrust force that propels the rocket forward. This thrust force is equal to the rate of change of momentum, which is the mass of the fuel times its velocity relative to the rocket. However, in order to maintain the overall momentum of the rocket-fuel system, the rocket itself must experience an equal and opposite change in momentum. This means that the velocity of the rocket will decrease by the same amount as the velocity of the fuel being ejected. Therefore, the velocity of the fuel relative to the Earth is v-v_e, as it is the difference between the velocity of the rocket (v) and the velocity of the fuel relative to the rocket (v_e). This relationship is essential in understanding and calculating the thrust and acceleration of a rocket during propulsion.