Rocket Propulsion

What is Rocket Propulsion?

Let us assume a rocket of total initial mass (rocket + fuel) m₀, starts moving upward due to the thrust force of the fuel jet. Assuming the velocity of the fuel jet with respect to the rocket to be u (assumed to be constant for this discussion) in a vertically downward direction and the mass of jet fuel emerging out of the rocket per unit time to be dmdt. Let the velocity of rocket after t time of motion be v and the acceleration of the rocket be a in vertically upward direction.

Thrust on the Rocket

F=−udmdt

Where F= Thrust
dmdt= rate of ejection of the fuel

u=velocity of exhaust gas with respect to rocket

m=mass of the rocket at time t

The net force on rocket-

Fnet =−udmdt−mg∗F=−udmdt[ if gravity neglected ]

Acceleration of Rocket (a)

a=−umdmdt−g

If g is neglected then
a=−umdmdt

Instantaneous Velocity of Rocket (v)

If g is neglected then-

a=−umdmdtdvdt=−umdmdt∫0vdv=−u∫m0m1mdm⇒v=uloge⁡(m0m)v=u∗loge⁡(m0m)=2.303u∗log10⁡(m0m)

Burnt Speed of Rocket

• It is the speed attained by the rocket when complete fuel gets burnt.

• It is the maximum speed attained by the rocket

• Formula

Vb=Vmax=uloge⁡(m0mr)
Vb→ burnt speed
mr→ residual mass of empty container

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Solved Example Based On Rocket Propulsion

Example 1: A rocket with a lift-off mass 3.5×104 kgis blasted upwards with an initial acceleration of 10 m/s2. Then the initial thrust of the blast is:

1) 3.5×105 N
2) 7.0×105 N
3) 14.0×105 N
4) 1.75×105 N

Solution:

Rocket Propulsion

dmdt= rate of ejection of the fuel
* Thrust on the rocket
F=−udmdt−mg
F= Thrust
∗F=−udmdt[ if gravity neglected ]

wherein

The negative sign indicates the direction of thrust is opposite to the direction of the gases.

Initial thrust = (Lift-off mass) $\times$ (a + g)

=(3.5×104)×(20)=7×105 N

Hence, the answer is option (2).

Example 2: A rocket of mass 1000 Kg set for vertical firing the exhaust speed is 200 m/s to give on initial acceleration of what will be the amount of gas ejected per second (in Kg/s ) to supply the needed thrust (g = 10 m/s )

1) 125

2) 100

3) 150

4) 75

Solution:

Acceleration of Rocket

a=umdmdt−ga=umdmdt [If g is neglected]

wherein

Instantaneous Velocity of Rocket

v=uloge⁡(m0m)−gt
m∘→ Initial mass of Rocket

Force applied by the gases on the rocket =vdmdt

vdmdt−mg=madmdt=m(g+a)v=1000(10+15)200=125Kg/s

Hence, the answer is option (1).

Example 3: If the maximum possible exhaust velocity of a rocket is 2 Km/s, calculate the ratio m0mr for it to require the escape velocity of 11.2 km/s after starting from rest (approx) [ m0→ initial velocity, mr→ mass of emptied rocket]

1) 270

2) 300

3) 350

4) 200

Solution:

Given

Maximum relative velocity of fuel, u=2000m/s,

Escape velocity of rocket v_b=11.2km/hr=11200m/s,

Neglecting the effect of gravity, the expression for the speed of rocket-

Vb=Vmax =uloge⁡(m0mr)mr→ residual mass of empty container Vb=2.303ulog⁡(m0mr)11.2=2.303×2×log10⁡(m0mr)log10⁡(m0mr)=11.22.303×2=2.432(m0m)=antilog(2.432)=270.4

Hence, the answer is the option (1).

Example 4: A spaceship in space sweeps stationary interplanetary dust. As a result, its mass increases at a rate dM(t)dt=bv2(t) where v(t) is its instantaneous velocity. The instantaneous acceleration of the satellite is:

1) −2bv3M(t)
2) −bv3M(t)
3) −bv3(t)
4) −bv32M(t)

Solution:

As given,

dM(t)dt=bv2(t)

As we know,

F=d(Mv)dt

So,
F=Mdvdt+vdMdtF=M(dvdt)+v(bv2)

We know that the net force is zero. F=0
dvdt=a=(−bv3M(t))

Where M(t) represents Mass as a function of times.

Hence, the answer is the option (2).

Example 5: The initial mass of the rocket is 1000 kg. Calculate at what rate the fuel should be burnt so that the rocket is given an acceleration of 20 ms^-2. The gases come out at a relative speed of [Useg=10 m/s2]

1) 10Kgs−1
2) 60Kgs−1
3) 500Kgs−1
4) 6⋅0×102Kgs−1

Solution:

Minitial =1000 kga=20ms2Vrel =500ms

F−mg=maVrel dmdt−mg=ma(500)dmdt−1000×10=1000500(dmdt)=1000(30)⇒(dmdt)=60(kgs)

Hence, the answer is the option (2).

Summary

Rocket propulsion is the application of Newton's third law. It works on the principle of momentum conservation. In a rocket, gases at high temperatures and pressure, are produced by the combustion of fuel. They escape with a large constant velocity through a nozzle.