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