Sheet 04 Variable Mass Situation: Rocket Propulsion.
A rocket burns \(0.5kg\) of fuel per second ejecting it as gases with a velocity of \(1600m.s^{-1}\) relative to rocket. How much force is exerted on the rocket? Also calculate: the velocity attained by rocket when its mass reduces to \(\left(\frac{1}{200}\right)^{th}\) of its initial mass.
Ans: \(F=800N\) and \(8.478\frac{km}{h}\).
A rocket has a mass of \(\left(2\times 10^{4}\right) kg\) of which half is fuel. Assume that the fuel is consumed at a constant rate as the rocket is fired and there is a constant thrust of \(\left(5\times 10^{6}\right)N\). Neglecting the air resistance and any possible variation of g, Compute: (a) Initial acceleration. (b) Acceleration when the whole fuel is consumed.
Ans:(a) \(a=240\frac{m}{s^{2}}\) and (b) \(a’=490\frac{m}{s^{2}}\).
Fuel is consumed at a rate of \(100 kg.s^{-1}\) in a rocket. The exhaust gases are ejected at a speed of \(\left(4.5\times 10^{4}\right) m.s^{-1}\). Calculate: what thrust is experienced by the rocket?
Ans: \(F=\left(4.5\times10^{6}\right)N\).
What should be the ratio \(\frac{m_{o}}{m}\) for a rocket if it is to escape from earth. Given: Escape velocity = \(11.2\frac{km}{s}\) and exhaust speed of gases is \(2\frac{km}{s}\).
Ans: \(\frac{m_{o}}{m}=270.4\).
A rocket of initial mass \(6000kg\) ejects mass at a constant rate of \(16kg.s^{-1}\) with constant relative speed of \(11 km.s^{-1}\). What is the acceleration of the rocket a minute after the blast? Neglect gravity.
Ans: \(a=34.92\frac{m}{s^{2}}\).
A rocket motor consumes \(100kg\) of fuel per second, exhausting at a rate of \(\left(4\times 10^{3}\right)m.s^{-1}\). (a) What force is exerted on the rocket? (b) What will be the velocity of the rocket at the instant its mass is reduced to \(\left(\frac{1}{30}\right)^{th}\) of the initial mass? Take: The initial velocity of the rocket as Zero and neglect gravity.
Ans:(a) \(F=\left(4\times10^{4}\right)N\) and (b) \(13.60\times10^{3}\frac{m}{s}\).
A rocket is set for vertical launch. If the exhaust speed is \(1000m.s^{-1}\). How much gas will be ejected per second to supply the thrust needed. (a) To overcome the weight of the rocket. (b) To give the rocket an initial vertical upwards acceleration of \(29.6m.s^{-2}\). Given: mass of the rocket is \(5000kg\).
Ans:(a) \(\frac{dm}{dt}=49\frac{kg}{Sec}\). and (b) \(\frac{dm}{dt}=197\frac{kg}{s}\)
A rocket fired from the earth’s surface ejects \(1\%\) of its mass at a speed of \(2000m.s^{-1}\) in the first second. Find: the average acceleration of the rocket in the first second?
Ans: \(a=20\frac{m}{s^{2}}\).
A balloon of mass m is rising up with an acceleration of a. Show that the faction of weight of the balloon that must be detached in order to double its acceleration is \(\frac{ma}{\left(2a+g\right)}\). Assume: the upthrust of air remains the same.
