Particles of mud are thrown from the rim of a rolling wheel. If the forward speed of the wheel is $v_{0}$, and the radius of the wheel is b, show that the greatest height above the ground that the mud can go is $b+\frac{v_{0}^{2}}{2 g}+\frac{g b^{2}}{2 v_{0}^{2}}$ At what point on the rolling wheel does this mud leave? (Note: It is necessary to assume that $v_{0}^{2} \geq b g$.)

Use the forms provided in the Working Papers. For the statement, prepare a flowchart to represent the process described.

A mailroom clerk receives customer checks, endorses the checks, and forwards the checks to the cashier. The cashier uses the checks to prepare a deposit slip.

A rocket coasts in an elliptical orbit around Earth. To attain the greatest amount of KE for escape for a given amount of fuel, should it fire its engines to accelerate forward when it is at the apogee or at the perigee? (Hint: Let the formula $F d=\Delta \mathrm{KE}$ be your guide to thinking. Suppose the thrust $F$ is brief and of the same duration in either case. Then consider the distance $d$ the rocket would travel during this brief burst at the apogee and at the perigee.)

The $K_{\mathrm{c}}$ at $100^{\circ} \mathrm{C}$ is $2.0$ for the decomposition reaction of NOBr. $2 \operatorname{NOBr}(g) \rightleftarrows 2 \mathrm{NO}(g)+\mathrm{Br}_2(g)$ $In an experiment,$1.0$mole of$ $\mathrm{NOBr}$, 1.0$mole of$ $\mathrm{NO}, and$1.0$mole of$ $\mathrm{Br}$_2$were placed in a$1.0 $$ \mathrm{~L}$ container. c. If not, will the rate of the forward or reverse reaction initially speed up? $$

Squids can move through the water using a form of jet propulsion. Suppose a squid jets forward from rest with constant acceleration for $0.170 \mathrm{~s}$, moving through a distance of $0.179 \mathrm{~m}$. The squid then turns off its jets and coasts to rest with constant acceleration. The total time for this motion (from rest to rest) is $0.400 \mathrm{~s}$, and the total distance covered is $0.421 \mathrm{~m}$. What is the time it coasts to a stop?

In which of the following situations is the moving object appropriately modeled as a projectile? Choose all correct answers.

(a) A shoe is tossed in an arbitrary direction.

(b) A jet airplane crosses the sky with its engines thrusting the plane forward.

(c) A rocket leaves the launch pad.

(d) A rocket moves through the sky, at much less than the speed of sound, after its fuel has been used up.

(e) A diver throws a stone underwater.

Use the forward-difference formulas and backward-difference formulas to determine each missing entry in the following tables. a.

$$\begin{matrix} \text{x} & \text{f(x)} & f^{\prime}(x)\\ \text{0.5} & \text{ 0.4794 }\\ \text{0.6} & \text{ 0.5646 }\\ \text{0.7} & \text{ 0.6442}\\ \end{matrix}$$

b.

$$\begin{matrix} \text{x} & \text{f(x)} & f^{\prime}(x)\\ \text{0.0} & \text{ 0.00000 }\\ \text{0.2} & \text{ 0.74140 }\\ \text{0.4} & \text{1.3718 }\\ \end{matrix}$$

A circular wheel with radius 1 ft and center C rolls to the right along the x-axis at a half-turn per second. (See the accompanying figure.) At time t seconds, the position vector of the point P on the wheel's circumference is

$$\mathbf{r}=(\pi t-\sin \pi t) \mathbf{i}+(1-\cos \pi t) \mathbf{j} .$$

At any given time, what is the forward speed of the topmost point of the wheel? Of C ?

The below statement corresponds to a numbered sentence in the passage. It contains a blank and is followed by four answer choices. Decide which choice fits best in the blank. The word or phrase that you choose must express roughly the same meaning as the italicized word in the passage. Write the letter of your choice on the answer line.

The Nile becomes $\underline{?}$ , almost losing its forward motion. ___________ (torpid)

a. destructive b. relentless c. renewed d. sluggish

Study the entries and answer the question that follows.

The roots grad and gress mean "step."

The roots ced and cess mean "go."

The prefix de- means "down."

The prefix pro- means "forward."

The prefix re- means "back."

The prefix di- means "apart" or "away from."

The prefix trans- means "across" or "through."

Using literal translations as guidance, define the following words without using a dictionary.

procession

If the 3-month forward exchange rate is USD1.55 = GBPI and the 3-month interest rate on dollars is 3.5% (effective annual rate), what do you think is the 3-month sterling (U.K.) interest rate? Explain what would happen if the rate were substantially above your figure. (Hint: In your calculations remember to convert the annually compounded interest rate into a rate for 3 months.)

Consider a common mirage formed by superheated air immediately above a roadway. A truck driver whose eyes are 2.00 m above the road, where n = 1.000293, looks forward. She perceives the illusion of a patch of water ahead on the road. The road appears wet only beyond a point on the road at which her line of sight makes an angle of $1.20^{\circ}$ below the horizontal. Find the index of refraction of the air immediately above the road surface.

Ammonia gas dissolves in water according to the following equation:

$$\mathrm{NH}_3(g)+\mathrm{H}_2 \mathrm{O}(l) \rightleftarrows \mathrm{NH}_4^{+}(a q)+\mathrm{OH}^{-}(a q)+\text { energy; } K=1.8 \times 10^{-5}$$

______________a. Is aqueous ammonia an acid or a base?

______________b. Is the equation given above an example of hydrolysis?

______________c. For the given value of $K$, does the equilibrium favor the forward or reverse reaction?

Consider a diff-amp, with parameters $V^{+}=5 \mathrm{~V}$, $V^{-}=-5 \mathrm{~V}$, and $I_Q=0.4 \mathrm{~mA}$. (a) Redesign the circuit such that the commonmode input voltage is in the range $-3 \leq v_{c m} \leq 3 \mathrm{~V}$, while $Q_1$ and $Q_2$ remain biased in the forward-active region. (b) Using the results of part (a), find the differentialmode voltage gain $A_d=\left(v_{c 2}-v_{c 1}\right) / v_d$.