Question

State the principle of rocket. With respect to rockets, define orbital velocity and escape velocity. Write the relationship between them.

Answer 1

HintRelate it to Newton’s third law of motion and conservation of momentum. Calculate orbital velocity and escape velocity using the concept of centripetal force. Orbiting velocity should be fast enough to counteract the force of gravity pulling the orbiting object towards the earth. Balance the centripetal and the gravitational force of earth to get orbital velocity. Escape velocity is greater than the orbital velocity since it is the velocity needed to break free from the orbits. To calculate the escape velocity you need to equate kinetic energy to the work done by force of gravity. You can find work done by the force of gravity by finding the product of gravitational force and the small displacement and then integrating it over the entire displacement to get the work done by the gravitational force of earth.

 Complete step-by-step solution:Rocket in its simplest form is a chamber enclosing gas under pressure. A small opening at one end of this chamber allows the gas to escape and in doing so provides the thrust that propels the rocket in the opposite direction. A good example of this is the balloon. Air inside the balloon is compressed by the balloon’s rubber walls. When the nozzle is released, air escapes through it and the balloon is propelled in the opposite direction.
Rocket principle can be easily explained using newton's law of motion. An unbalanced force must be exerted for the rocket to lift off from the Launch pad (first law).
The amount of thrust, force produced by the rocket engine, will be determined by the mass of the rocket fuel that is burnt and how fast the gas escapes the rocket (second law).
The reaction or motion of the rocket is equal to and in the opposite direction of the action, or thrust from the engine (third law).
Now we will define the orbital velocity.
Orbital velocity is the speed required to achieve the orbit around the celestial body such as a planet or the star. Orbital velocity can be derived by equating the gravitational and the centripetal force.
Gravitational force between earth and rocket,
${F_g} = \dfrac{{GMm}}{{{r^2}}}$
Centripetal force,
${F_c} = \dfrac{{m{V^2}}}{r}$
$G$is the gravitational constant,
$M$is the mass of earth,
$m$is mass of the rocket,
$V$is the velocity of rocket in the orbit,
$r$is the radius of the orbit.
Now $r = R + h$
$R = $radius of the earth
$h = $height of the rocket
Balancing both the forces, we have:
${F_g}$$ = {F_c}$
$V = \sqrt {gr} $
This gives us the orbital velocity.
Escape velocity is the velocity required to break free this orbit and escape the gravitational field of the earth.
Escape velocity ${V_e} = \sqrt {2gr} $
Relation between ${V_e},{V_o}$
${V_e} = \sqrt 2 {V_o}$

Note: To calculate these relations, we must properly know the definitions of each term. Try relating all the terms to real life situations and draw the parallel to understand them better.