Motion of a System with Varying Mass: Rocket

. Rocket

A rocket employs the principle of jet propulsion. It may be a missile, spacecraft, or other vehicle that obtains thrust from a rocket engine. The exhaust of a rocket engine is produced entirely from propellants carried within the rocket before its launch.

The operation of a rocket engine is based on Newton's third law of motion and the law of conservation of linear momentum. A rocket is propelled forward by ejecting exhaust gases backward at a very high velocity.

A rocket engine consists essentially of propellant tanks, a combustion chamber, and a nozzle. The propellants may be gaseous, solid, liquid, or a combination of solid and liquid propellants.

In the combustion chamber, a chemical reaction takes place between the fuel and the oxidizer, producing gases at extremely high temperature and pressure. The gases escape through a nozzle at very high velocity. The backward ejection of these gases produces an equal and opposite reactive thrust on the rocket, according to Newton's third law.

-----------------------------------

Let a system of mass \(M_0\) start from level A at a time \(t_0\).

The velocity of a system of mass \(M\), at any instant of time \(t\), be \(\vec{v}\) at level B.

Suppose that during the time interval at level C from \(t\) to \(t+d t\), a small mass \(d M\) is separated from the system and moves with velocity \(\vec{u}\). After separation, the remaining part of the system has mass \(M-d M\) and moves with velocity \(\vec{v}+d\vec{v}\).

If the separated mass \(d M\) and the remaining mass \(M-d M\) are considered together as the same system, then, according to Newton's second law between level B and C,

\[ \vec{F}_{\mathrm{ext}} = \frac{\vec{P}_c-\vec{P}_b}{d t} \]

Therefore,

\[ \vec{F}_{\mathrm{ext}} = \frac{ [(M-d M)(\vec{v}+d\vec{v}) +d M\,\vec{u}] -M\vec{v} }{d t}. \] \[ \vec{F}_{\mathrm{ext}} = \frac{ M\vec{v}-d M\vec{v}+M\,d\vec{v}-dM\,d\vec{v} +d M\,\vec{u} -M\vec{v} }{d t}. \]

On expanding and neglecting the product \(d M\,d\vec{v}\), which is a second-order small quantity, we obtain

\[ \boxed{ \vec{F}_{\mathrm{ext}} = M\frac{d\vec{v}}{dt} + (\vec{u}-\vec{v}) \frac{dM}{dt} } \]

(1)

Rearranging Eq. (1),

\[ M\frac{d\vec{v}}{dt} = \vec{F}_{\mathrm{ext}} + (\vec{u}-\vec{v})\frac{dM}{dt}. \]

The relative velocity of the separated mass with respect to the system is

\[ \boxed{ \vec{u}_r=\vec{u}-\vec{v} } \]

Hence,

\[ \boxed{ M\frac{d\vec{v}}{dt} = \vec{F}_{\mathrm{ext}} + \vec{u}_r\frac{dM}{dt} } \]

(2)

If the change in gravitational force with height and air resistance are assumed to be negligible, the external force acting on the rocket is its weight:

\[ \vec{F}_{\mathrm{ext}}=M\vec{g}. \]

Therefore, Eq. (1) becomes

\[ M\frac{d\vec{v}}{dt} = M\vec{g} + \vec{u}_r\frac{dM}{dt}. \]

(2)

For an upward-moving rocket, both gravitational acceleration \(\vec{g}\) and the relative exhaust velocity are directed downward. Taking the upward direction as positive, the equation becomes

\[ M\frac{dv}{dt} = -Mg - u_r\frac{dM}{dt}. \]

Dividing by \(M\),

\[ \frac{dv}{dt} = -g - u_r\frac{1}{M}\frac{dM}{dt}. \]

Hence,

\[ dv=-g\,dt-u_r\frac{dM}{M}. \]

(3)

Integrating Eq. (3), from level A to B

\[ \int_{v_0}^{v} dv = -g\int_{0}^{t} dt - u_r\int_{M_0}^{M}\frac{dM}{M}. \]

Therefore,

\[ v-v_0=-gt-u_r(\ln M-\ln M_0), \] \[ v=v_0+u_r(\ln M_0-\ln M)-gt, \]

(4)

Using the logarithmic identity, we obtain

\[ \boxed{ v = v_0 + u_r\ln\left(\frac{M_0}{M}\right) - gt } \]

(5)

This is known as a Rocket Equation

Special Case: No Gravitational Force

If gravitational force is neglected, \(g=0\), and the rocket equation reduces to

\[ \boxed{ v=v_0 + u_r\ln\left(\frac{M_0}{M}\right) } \]

This is the fundamental Tsiolkovsky rocket equation. It shows that the change in velocity of a rocket depends on the relative velocity of the exhaust gases and the ratio of the initial mass to the instantaneous mass of the rocket.

Multistage Rocket

A multistage rocket is a rocket made up of two or more stages, arranged one above another. Each stage contains its own fuel, oxidizer, combustion chamber, and rocket engine. After the propellant of a stage is exhausted, that stage is separated (discarded) from the rocket, reducing the total mass that the remaining stages have to accelerate.

The working of a multistage rocket is based on the conservation of linear momentum and Newton's third law of motion.

When the rocket ejects gases backward with high velocity, the rocket receives an equal and opposite forward momentum, producing thrust.