Find the velocity of the aerosol solid particles when $t=10\ \mu \mathrm{s}$, if when $t=0$ they leave the can with a horizontal velocity of $30 \mathrm{~m} / \mathrm{s}$. Assume the average diameter of the particles is $0.4\ \mu \mathrm{m}$ and each has a mass of $0.4\left(10^{-12}\right) \mathrm{g}$. The air is at $20^{\circ} \mathrm{C}$. Neglect the vertical component of the velocity. Note: The volume of a sphere is $t=\frac{4}{3} \pi r^3$.

Verify that the incompressible flow distribution, in cylindrical coordinates,

$$v_r=0 \quad v_\theta=C r^n \quad v_z=0$$

where $C$ is a constant, (a) satisfies the Navier-Stokes equation for only two values of $n$. Neglect gravity. (b) Knowing that $p=p(r)$ only, determine the pressure distribution for each case, assuming that the pressure at $r=R$ is $p_0$. What might these two cases represent?

A revolving searchlight, which is 100m from the nearest point on a straight highway, casts a horizontal beam along a highway. The beam leaves the spotlight at an angle of $\pi / 16 \mathrm{rad}$ and revolves at a rate $\pi / 6 \mathrm{rad} / \mathrm{s}$. Let $w$ be the width of the beam as it sweeps along the highway and let $\theta$ be the angle that the center of the beam makes perpendicular to the highway. What is the rate of change of $w$ when $\theta=\pi / 3$? Neglect the height of the searchlight.

A steam turbine operates steadily under the following conditions. At the inlet, p = 2.5 MPa, T = 450°C, and V = 40 m/s. At the outlet, p = 22 kPa, T = 70°C, and V = 225 m/s. (a) If we neglect elevation changes and heat transfer, how much work is delivered to the turbine blades, in kJ/kg? (b) If the mass flow is 10 kg/s, how much total power is delivered? (c) Is the steam wet as it leaves the exit?

Steep safety ramps are built beside mountain highways to enable vehicles with defective brakes to stop. A $10$-ton truck enters a $15^{\circ}$ ramp at a high speed $v_0=108 \mathrm{ft} / \mathrm{s}$ and travels for $6 \mathrm{~s}$ before its speed is reduced to $36 \mathrm{ft} / \mathrm{s}$. Assuming constant deceleration, determine $(a)$ the magnitude of the braking force, $(b)$ the additional time required for the truck to stop. Neglect air resistance and rolling resistance.

Neglect all forces except gravity. In all these situations, the effect of air resistance is actually significant, but your calculations will give a good first approximation. A baseball pitcher throws a pitch horizontally from a height of 6 feet with an initial speed of 130 feet per second. If a person drops a ball from height 6 feet at the same time the pitcher releases the ball, how high will the dropped ball be when the pitch crosses home plate?

A prototype fan has a 20 -ft diameter, an inlet pressure of 14.40 psia, an inlet temperature of 70$^{\circ}$ F, and a speed of 90 rpm. A 1/10-scale model of the fan has the same inlet pressure and temperature, an inlet power of $1.24 \mathrm{hp}$, a flow rate of $220 \mathrm{ft}^3 / \mathrm{min}$, and a speed of $1800 \mathrm{rpm}$. Find the corresponding input power and flow rate of the prototype fan. Neglect Reynolds number effects.

A projectile is fired at $35^{\circ}$ above the horizontal. At the highest point in its trajectory, its speed is $200 \mathrm{~m} / \mathrm{s}$. The initial velocity had a horizontal component of $(a) 0$; (b) $(200 \mathrm{~m} / \mathrm{s}) \cos 35^{\circ} ;(c)(200 \mathrm{~m} / \mathrm{s}) \sin 35^{\circ} ;(d)(200 \mathrm{~m} / \mathrm{s}) /$ $\cos 35^{\circ} ;(e) 200 \mathrm{~m} / \mathrm{s}$. Neglect the effects of air resistance.

At a certain temperature, $K = 1.1 \times 10 ^ { 3 }$ for the reaction

$$\mathrm { Fe } ^ { 3 + } ( a q ) + \mathrm { SCN } ^ { - } ( a q ) \rightleftharpoons \mathrm { FeSCN } ^ { 2 + } ( a q )$$

Calculate the concentrations of $\mathrm { Fe } ^ { 3 + } ,\ \mathrm { SCN } ^ { - } ,$ and $\mathrm { FeSCN } ^ { 2 + }$ at equilibrium if 0.020 mole of $\mathrm { Fe } \left( \mathrm { NO } _ { 3 } \right) _ { 3 }$ is added to 1.0 L of 0.10 M KSCN. (Neglect any volume change.)

A compressor receives $0.05 \mathrm{~kg} / \mathrm{s} \mathrm{R}-410 \mathrm{a}$ at $200 \mathrm{kPa}$, $-20^{\circ} \mathrm{C}$ and $0.1 \mathrm{~kg} / \mathrm{s} \mathrm{R}-410 \mathrm{a}$ at $400 \mathrm{kPa}, 0^{\circ} \mathrm{C}$. The exit flow is at $1000 \mathrm{kPa}, 60^{\circ} \mathrm{C}$, as shown in figure. Assume it is adiabatic, neglect kinetic energies, and find the required power input.

Neglect all forces except gravity. In all these situations, the effect of air resistance is actually significant, but your calculations will give a good first approximation. A baseball pitcher throws a pitch horizontally from a height of 6 feet with an initial speed of 130 feet per second. Find a vector-valued function describing the position of the ball t seconds after release.

An object is projected vertically from the surface of Earth at less than the escape speed. Show that the maximum height reached by the object is $H=R_{\text{E}} H^{\prime} /\left(R_{\text{E}}-H^{\prime}\right)$, where $H^{\prime}$ is the height that it would reach if the gravitational field were constant. Neglect any effects due to air resistance.

A 200-1b force applied to the end of the piston of the shock absorber shown in Fig. P5. 136 causes the two ends of the shock absorber to move toward each other with a speed of $5\ \mathrm{ft} / \mathrm{s}$. Determine the head loss associated with the flow of the oil through the channel. Neglect gravity and any friction force between the piston and cylinder walls.

At a certain temperature, $K=1.1 \times 10^{3}$ for the reaction

$$\mathrm{Fe}^{3+}(a q)+\mathrm{SCN}^{-}(a q) \rightleftharpoons \mathrm{FeSCN}^{2+}(a q)$$

Calculate the concentrations of $\mathrm{Fe}^{3+}, \mathrm{SCN}^{-},$ and $\mathrm{FeSCN}^{2+}$ at equilibrium if 0.020 mole of $\mathrm{Fe}\left(\mathrm{NO}_{3}\right)_{3}$ is added to 1.0 L of 0.10 M KSCN. (Neglect any volume change.)