Variable Mass System

This problem is designed to illustrate the advantage that can be obtained by the use of multiple-staged instead of single-staged rockets as launching vehicles. Suppose that the payload (e.g., a space capsule) has mass $m$ and is mounted on a two-stage rocket (see figure). The total mass (both rockets fully fuelled, plus the payload) is$\mathrm{Nm}.$

The mass of the second-stage rocket plus the payload, after first-stage burnout and separation, is $nm$. In each stage the ratio of container mass to initial mass (container plus fuel) is $r,$ and the exhaust speed is $V,$ constant relative to the engine. Note that at the end of each state when the fuel is completely exhausted, the container drops off immediately without affecting the velocity of rocket. Ignore gravity.

$\left(\mathrm{a}\right)$ Obtain the velocity $v$ of the rocket gained from the first-stage burn, starting from rest in terms of $\left\{V,N,n,r\right\}$

$\left(\mathrm{b}\right)$ Obtain a corresponding expression for the additional velocity $u$ gained from the second stage burn.

$\left(\mathrm{c}\right)$Adding$v\text{ and} u$, you have the payload velocity $w$ in terms of $N, n,\text{ and} r.$Taking $N$ and $r$ as constants, find the value of $n$ for which $w$ is a maximum. For this maximum condition obtain $\frac{u}{v}.$

$\left(\mathrm{d}\right)$ Find an expression for the payload velocity $w_{\mathrm{s}}$of a single-stage rocket with the same values of $N, r,\text{ and} V$

$\left(\mathrm{e}\right)$Suppose that it is desired to obtain a payload velocity of $10 \mathrm{km} {\mathrm{s}}^{-1},$ using rockets which $V=2.5 \mathrm{km} {\mathrm{s}}^{-1}$ and $r=0.1$. Using the maximum condition of part $\left(\mathrm{c}\right)$ obtain the value of $N$ if the job is to be done with a two-stage rocket.

A spaceship of mass $m_0$ moves in the absence of external forces with a constant velocity $v_0.$ To change the motion direction a jet engine is switched on. It starts ejecting a gas jet with velocity $u,$ which is constant relative to the spaceship and directed at right angles to the spaceship motion. The engine is shut down when the mass of the spaceship decreases to m. Through what angle $\mathrm{\alpha }$ did the motion direction of the spaceship deviate due to the jet engine operation?

A uniform chain of mass $\mathrm{m} = 1\mathrm{kg}$ and length $l = 1\mathrm{m}$ hangs on a thread and touches the surface of a table by its lower end. Find the force exerted by the table on the chain when $\mathrm{x} = 0.5 \mathrm{m}$ length of chain has fallen on the table. The fallen part does not form heap.

If the thrust force on a rocket is ejecting gases with a relative velocity of $300 \mathrm{m} {\mathrm{s}}^{-1},$ is $210 \text{N}$. Then the rate of combustion of the fuel will be:

A rocket of mass, $m=20 \mathrm{kg}$ has $M=180 \mathrm{kg} \mathrm{fuel}.$ The uniform exhaust velocity of the fuel is $v_{\mathit{r}}= 1.6 \mathrm{km} {\mathrm{s}}^{-1}.$
(i) Calculate the minimum rate of consumption of fuel so that the rocket may rise from the ground.
(ii) Also calculate the maximum vertical speed gained by the rocket when the rate of consumption of fuel $\mathrm{\mu }$ is

$$\begin{gathered} \left(\mathrm{a}\right) 2 \mathrm{kg} {\mathrm{s}}^{-1} \\ \left(\mathrm{b}\right) 20 \mathrm{kg} {\mathrm{s}}^{-1} \end{gathered}$$

Use $(g\mathit{ }= 10 \mathrm{m} {\mathrm{s}}^{-2})$ and $(\ln 10 = 2.30)$