Why is the following situation impossible? An air rifle is used to shoot 1.00-g particles at a speed of vx = 100 m/s. The rifle’s barrel has a diameter of 2.00 mm. The rifle is mounted on a perfectly rigid support so that it is fired in exactly the same way each time. Because of the uncertainty principle, however, after many firings, the diameter of the spray of pellets on a paper target is 1.00 cm.

Dimensions of a centrifugal pump impeller are

$$\begin{matrix} \text{Parameter} & \text{Inlet, Section 1} & \text{Outlet, Section 2}\\ \text{Radius, r (mm)} & \text{175} & \text{500}\\ \text{Blade width, b (mm)} & \text{50} & \text{30}\\ \text{Blade angle, }{\beta (deg)} & \text{65} & \text{70}\\ \end{matrix}$$

The pump handles water and is driven at 750 rpm. Calculate the theoretical head and mechanical power input if the flow rate is 0.75 $\mathrm{m}^{3}$/s. For the impeller, operating at 750 rpm, determine the volume flow rate for which the tangential component of the inlet velocity is zero. Calculate the theoretical head and mechanical power input.

An object disintegrates into two fragments. One fragment has mass 1.00 MeV/c2 and momentum 1.75 MeV/c in the positive x direction, and the other has mass 1.50 MeV/c2 and momentum 2.00 MeV/c in the positive y direction. Find (a) the mass and (b) the speed of the original object.

Consider a flat, circular current loop of radius R carrying a current I. Choose the x axis to be along the axis of the loop, with the origin at the loop’s center. Plot a graph of the ratio of the magnitude of the magnetic field at coordinate x to that at the origin for x = 0 to x = 5R. It may be helpful to use a programmable calculator or a computer to solve this problem.

Suppose that the distribution of the number of items x produced by an assembly line during an 8-hour shift can be approximated by a normal distribution with mean value 150 and standard deviation 10. a. What is the probability that the number of items produced is at most 120? b. What is the probability that at least 125 items are produced? c. What is the probability that between 135 and 160 (inclusive) items are produced?

The intensity of solar radiation at the top of the Earth’s atmosphere is 1 370 W/m2. Assuming 60% of the incoming solar energy reaches the Earth’s surface and you absorb 50% of the incident energy, make an order-of-magnitude estimate of the amount of solar energy you absorb if you sunbathe for 60 minutes.

Because of safety considerations, in May 2003 the Federal Aviation Administration (FAA) changed its guidelines for how small commuter airlines must estimate passenger weights. Under the old rule, airlines used 180 pounds as a typical passenger weight (including carry-on luggage) in warm months and 185 pounds as a typical weight in cold months. The Alaska Journal of Commerce (May 25, 2003) reported that Frontier Airlines conducted a study to estimate average passenger plus carry-on weights. They found an average summer weight of 183 pounds and a winter average of 190 pounds. Suppose that each of these estimates was based on a random sample of 100 passengers and that the sample standard deviations were 20 pounds for the summer weights and 23 pounds for the winter weights. a. Construct and interpret a $95\%$ confidence interval for the mean summer weight (including carry-on luggage) of Frontier Airlines passengers. b. Construct and interpret a $95\%$ confidence interval for the mean winter weight (including carry-on luggage) of Frontier Airlines passengers. c. The new FAA recommendations are 190 pounds for summer and 195 pounds for winter. Comment on these recommendations in light of the confidence interval estimates from Parts (a) and (b).

Use the Integral Test to determine whether each series converges or diverges. $\sum_{k=1}^{\infty} \frac{1}{k^{2}+1}$

A sample of 548 ethnically diverse students from Massachusetts were followed over a 19-month period from 1995 and 1997 in a study of the relationship between TV viewing and eating habits (Pediatrics [2003]: 1321–1326). For each additional hour of television viewed per day, the number of fruit and vegetable servings per day was found to decrease on average by 0.14 serving. a. For this study, what is the dependent variable? What is the predictor variable? b. Would the least-squares line for predicting number of servings of fruits and vegetables using number of hours spent watching TV as a predictor have a positive or negative slope? Explain.

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.)

An incompressible liquid with negligible viscosity and density $\rho=1250 \mathrm{kg} / \mathrm{m}^{3}$ flows steadily through a 5-m-long convergent-divergent section of pipe for which the area varies as $A(x)=A_{0}\left(1+e^{-x / a}-e^{-x / 2 a}\right)$ where $A_{0}=0.25 \mathrm{m}^{2}$ and a=1.5 m. Plot the area for the first 5 m. Develop an expression for and plot the pressure gradient and pressure versus position along the pipe, for the first 5 m, if the inlet centerline velocity is 10 m/s and inlet pressure is 300 kPa. Hint: Use relation $u \frac{\partial u}{\partial x}=\frac{1}{2} \frac{\partial}{\partial x}\left(u^{2}\right)$.

Consider the velocity field $\vec{V}=A\left[x^{4}-6 x^{2} y^{2}+y^{4}\right] \hat{i}+B\left[x^{3} y-x y^{3}\right] \hat{j}$; $A=2 \mathrm{m}^{-3} \cdot \mathrm{s}^{-1}$, B is a constant, and the coordinates are measured in meters. Find B for this to be an incompressible flow. Obtain the equation of the streamline through point (x, y)=(1, 2). Derive an algebraic expression for the acceleration of a fluid particle. Estimate the radius of curvature of the streamline at (x, y)=(1, 2).

Show that the nth derivative of $\left(a x^{2}+b x+c\right) e^{x}$ is a function of the same form but with different constants.

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.