Question

A rocket of mass 1000 tons has an upward acceleration equal to 0.5 g How many kilogram of fuel must be
ejected per second at a relative speed of 2000 ms to produce the desired acceleration? (Ams: 7350 kg 1
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Answer 1

Momentum,

p = m × v

Now, force acting on a body is the rate of change of momentum.

So, F = dp / dt.

Thus, F = m × (dv / dt ) + v × ( dm / dt )

Now, assuming the rocket to be moving with a constant velocity,

dv / dt = 0

So, our equation reduces to,

F = v × ( dm / dt )

Now, rate of ejection of mass ( dm / dt ) = 50 kg / s

And, v = 20 m / s

So force acting on the rocket

= 20 × 50

= 1000 N

Now, effective mass of the rocket after 2 s ( M )

= 2000 - 2 × 50

= 1900 kg

So, the acceleration of the rocket

= F / M

= 1000 / 1900

= 0.526 m / s^2

Answer 2

According to the definition of Force:

Force is equal to the rate of change of momentum.

F = d(mv)/dt = m(dv/dt) + v(dm/dt)

Generally, the mass remains constant so we consider only one term [F=ma]!

In case of the rockets, the force of gravity is overcome by the force exerted by the emission of gases which helps the rocket to move upwards. Consequently, the mass keeps on varying and hence the second term needs to be considered as well! Therefore:

F = m(dv/dt) + v(dm/dt); where:

F = (mass of the rocket at that instant)*(acceleration experienced at that instant)

m(dv/dt) = External force acting on the object

v(dm/dt) = Force due to the emission of the gases

According to the problem;

m(dv/dt) = 0 {Gravity free space}

dm/dt = 50 kg/sec

m (at t=2 sec) = 2000-(50*2) = 1900 kg

v = 20 m/sec

m (at t=2 sec)*a = v(dm/dt)

1900a = 20*50

a = 0.52632 m/s^2