A $\mathrm{CO}_{2}$ cartridge is used to propel a small rocket cart. Compressed gas, stored at 35 MPa and $20^{\circ} \mathrm{C}$, is expanded through a smoothly contoured converging nozzle with 0.5 mm throat diameter. The back pressure is atmospheric. Calculate the pressure at the nozzle throat. Evaluate the mass flow rate of carbon dioxide through the nozzle. Determine the thrust available to propel the cart. How much would the thrust increase if a diverging section were added to the nozzle to expand the gas to atmospheric pressure? What is the exit area? Show stagnation states, static states, and the processes on a Ts diagram. Consider the converging-diverging option. To what pressure would the compressed gas need to be raised (keeping the temperature at $20^{\circ} \mathrm{C}$) to develop a thrust of 15N? (Assume isentropic flow.)

The following table gives the total world emissions of $\mathrm{CO}_{2}$ from fossil fuels, in billions of tons per year.

$$\begin{array}{c|c|c|c|c|c|c|} \hline {\text { Year }} & {1981} & {1986} & {1991} & {1996} & {2001} & {2006} & {2011} \\ \hline \mathrm{CO}_{2(\mathrm{bn}\text { tons per year) } } & {20.1} & {21.3} & {21.9} & {23.4} & {24.2} & {29.1} & {34.7} \\ \hline \end{array}$$

Use this data to estimate the total world $CO_2$ emissions between 1981 and 2011 using a right sum with n=3.

An insulation company is examining a new material for extruding into cavities. The experimental data is given below for the speed U of the upper plate, which is separated from a fixed lower plate by a 1-mm-thick sample of the material, when a given shear stress is applied. Determine the type of material. If a replacement material with a minimum yield stress of 250 Pa is needed, what viscosity will the material need to have the same behavior as the current material at a shear stress of 450 Pa?

$$\begin{matrix} \text{}{\tau(\mathrm{Pa})} & \text{50} & \text{100} & \text{150} & \text{163} & \text{171} & \text{170} & \text{202} & \text{246} & \text{349} & \text{444}\\ \text{U (m/s)} & \text{0} & \text{0} & \text{0} & \text{0.005} & \text{0.01} & \text{0.025} & \text{0.05} & \text{0.1} & \text{0.2} & \text{0.3}\\ \end{matrix}$$

For the following two-dimensional partial differential equations, find the dispersion relation: $(\mathrm{a})\ \frac{\partial^{2} u}{\partial t^{2}}=c^{2}\left(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}\right); (\mathrm{b})\ \frac{\partial^{2} u}{\partial t^{2}}=c_{1}^{2} \frac{\partial^{2} u}{\partial x^{2}}+c_{2}^{2} \frac{\partial^{2} u}{\partial y^{2}}.$

Compute the convolution of each of the following pairs of functions. $\sin a t, \sin b t, a \neq b$

Show that $\int_{e}^{\infty} \frac{d x}{x(\ln x)^{p}}$ converges if p>1 and diverges if $p \leq 1$.

Two vibrations at right angles to one another are described by the equations

$$\begin{array}{l} x=10 \cos (5 \pi t) \\ y=10 \cos (10 \pi t+\pi / 3) \end{array}$$

Construct the Lissajous figure of the combined motion.

Determine whether each improper integral converges or diverges. If it converges, find its value. $\int_{-\infty}^{1} \frac{x d x}{\sqrt{2-x}}$

Air enters a converging–diverging nozzle of a supersonic wind tunnel at 1 MPa and 300 K with a low velocity. If a normal shock wave occurs at the exit plane of the nozzle at $M_a =2.4$, determine the pressure, temperature, Mach number, velocity, and stagnation pressure after the shock wave.

Air flows isentropically through a converging nozzle into a receiver where the pressure is 250 kPa (abs). If the pressure is 350 kPa (abs) and the speed is 150 m/s at the nozzle location where the Mach number is 0.5, determine the pressure, speed, and Mach number at the nozzle throat.

Plot the points and find the slope of the line passing through the points. (-1, 5), (2, -3).

An object of mass m is projected vertically downward with initial velocity $V_0$ in a medium offering resistance proportional to the square root of the magnitude of the velocity. (a) Find a relation between the velocity V and the time t if the drag force equals $c \sqrt{V}$ (b) Find the terminal velocity of the object.

Find each definite integral. $\int_{0}^{2} \frac{x^{2}}{\sqrt{9+x^{2}}} d x$

Explain how to determine whether two nonzero vectors u and v are parallel.