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A 60.0-kg and a 90.0-kg skydiver jump from an airplane at an altitude of $6.00\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{3}\text{m}$ , both falling in the pike position. Make some assumption on their frontal areas and calculate their terminal velocities. How long will it take for each skydiver to reach the ground (assuming the time to reach terminal velocity is small)? Assume all values are accurate to three significant digits.
A 560-g squirrel with a surface area of $930\phantom{\rule{0.2em}{0ex}}{\text{cm}}^{2}$ falls from a 5.0-m tree to the ground. Estimate its terminal velocity. (Use a drag coefficient for a horizontal skydiver.) What will be the velocity of a 56-kg person hitting the ground, assuming no drag contribution in such a short distance?
${v}_{\text{T}}=25\phantom{\rule{0.2em}{0ex}}\text{m/s;}{\text{v}}_{2}=9.9\phantom{\rule{0.2em}{0ex}}\text{m/s}$
To maintain a constant speed, the force provided by a car’s engine must equal the drag force plus the force of friction of the road (the rolling resistance). (a) What are the drag forces at 70 km/h and 100 km/h for a Toyota Camry? (Drag area is $0.70\phantom{\rule{0.2em}{0ex}}{\text{m}}^{2}$ ) (b) What is the drag force at 70 km/h and 100 km/h for a Hummer H2? (Drag area is $2.44\phantom{\rule{0.2em}{0ex}}{\text{m}}^{2})$ Assume all values are accurate to three significant digits.
By what factor does the drag force on a car increase as it goes from 65 to 110 km/h?
${\left(\frac{110}{65}\right)}^{2}=2.86$ times
Calculate the velocity a spherical rain drop would achieve falling from 5.00 km (a) in the absence of air drag (b) with air drag. Take the size across of the drop to be 4 mm, the density to be $1.00\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{3}\phantom{\rule{0.2em}{0ex}}{\text{kg/m}}^{3}$ , and the surface area to be $\pi {r}^{2}$ .
Using Stokes’ law, verify that the units for viscosity are kilograms per meter per second.
Stokes’ law is ${F}_{\text{s}}=6\pi r\eta v.$ Solving for the viscosity, $\eta =\frac{{F}_{\text{s}}}{6\pi rv}.$ Considering only the units, this becomes $\left[\eta \right]=\frac{\text{kg}}{\text{m}\xb7\text{s}}.$
Find the terminal velocity of a spherical bacterium (diameter $2.00\phantom{\rule{0.2em}{0ex}}\text{\mu m}$ ) falling in water. You will first need to note that the drag force is equal to the weight at terminal velocity. Take the density of the bacterium to be $1.10\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{3}\phantom{\rule{0.2em}{0ex}}{\text{kg/m}}^{3}$ .
Stokes’ law describes sedimentation of particles in liquids and can be used to measure viscosity. Particles in liquids achieve terminal velocity quickly. One can measure the time it takes for a particle to fall a certain distance and then use Stokes’ law to calculate the viscosity of the liquid. Suppose a steel ball bearing (density $7.8\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{3}\phantom{\rule{0.2em}{0ex}}{\text{kg/m}}^{3}$ , diameter 3.0 mm) is dropped in a container of motor oil. It takes 12 s to fall a distance of 0.60 m. Calculate the viscosity of the oil.
$0.76\phantom{\rule{0.2em}{0ex}}\text{kg/m}\xb7\text{s}$
Suppose that the resistive force of the air on a skydiver can be approximated by $f=\text{\u2212}b{v}^{2}.$ If the terminal velocity of a 50.0-kg skydiver is 60.0 m/s, what is the value of b ?
A small diamond of mass 10.0 g drops from a swimmer’s earring and falls through the water, reaching a terminal velocity of 2.0 m/s. (a) Assuming the frictional force on the diamond obeys $f=\text{\u2212}bv,$ what is b ? (b) How far does the diamond fall before it reaches 90 percent of its terminal speed?
a. 0.049 kg/s; b. 0.57 m
(a) What is the final velocity of a car originally traveling at 50.0 km/h that decelerates at a rate of $0.400\phantom{\rule{0.2em}{0ex}}{\text{m/s}}^{2}$ for 50.0 s? Assume a coefficient of friction of 1.0. (b) What is unreasonable about the result? (c) Which premise is unreasonable, or which premises are inconsistent?
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