Will the rocket rise with your speed ?

Classical Mechanics Level 2

A rocket with initial mass $m$ discharges a jet of hot gases of mean density $\rho$ and effective area $A$ . The exhaust has a velocity of $u$ relative to the rocket.

What is the minimum value of $u$ that enables the rocket to rise vertically?

\sqrt{\dfrac{\rho gA}{m}}

\sqrt{\dfrac{\rho g}{mA}}

\sqrt{\dfrac{2mgA}{\rho}}

\sqrt{\dfrac{mg}{\rho A}}

1 solution

Steven Chase
Dec 9, 2017

Note: I would argue that this actually doesn't depend on $V$ (the rocket velocity), but rather depends on what I will call $u$ , which is the exhaust velocity relative to the rocket itself. With that said, here is my approach.

Incremental change in rocket momentum (here, $\rho$ is density, $p$ is momentum, $m$ is the rocket mass, and $M$ is the cumulative mass of expelled propellant):

$dp = dM \, u \\ \frac{dp}{dt} = F = u \frac{dM}{dt}$

Relate the exhaust mass flow rate to the density and the effective area:

$\frac{dM}{dt} = \rho A u \\ F = \rho A u^2$

Set the propulsion force expression equal to the gravitational force:

$\rho A u^2 = m g \\ u = \sqrt{\frac{m g}{\rho A}}$

True. Also we can apply the formula for Escape velocity which will easily give us the answer.

Kumudesh Ghosh - 1 year, 2 months ago

Will the rocket rise with your speed ?

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