6.4 Drag Force and Terminal Speed

Find the terminal velocity of a 50-kg skydiver falling in spread-eagle fashion.

Suppose the resistive force of the air on a skydiver can be approximated by $f = − b v^{2}$. If the terminal velocity of a 100-kg skydiver is 60 m/s, what is the value of b?

Athletes such as swimmers and bicyclists wear body suits in competition. Formulate a list of pros and cons of such suits.

As cars travel, oil and gasoline leaks onto the road surface. If a light rain falls, what does this do to the control of the car? Does a heavy rain make any difference?

The terminal velocity of a person falling in air depends upon the weight and the area of the person facing the fluid. Find the terminal velocity (in meters per second and kilometers per hour) of an 80.0-kg skydiver falling in a headfirst position with a surface area of $0.140 \text{m}^{2}$.

A 560-g squirrel with a surface area of $650 \text{cm}^{2}$ falls from a 5.0-m tree to the ground. Estimate its terminal velocity. (Use a drag coefficient for a skydiver falling feet first.) What will be the velocity of a 56-kg person hitting the ground, assuming no drag contribution in such a short distance?

By what factor does the drag force on a car increase as it goes from 65 to 110 km/h?

Using Stokes’ law, verify that the units for viscosity are kilograms per meter per second.

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 $7.8 \times 10^{3} \text{kg}/\text{m}^{3}$, diameter 3.0 mm) is dropped in a container of motor oil. It ta…

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 $f = − b v ,$ what is b? (b) How far does the diamond fall before it reaches 90 percent of its terminal speed?

A 75.0-kg woman stands on a bathroom scale in an elevator that accelerates from rest to 30.0 m/s in 2.00 s. (a) Calculate the scale reading in newtons and compare it with her weight. (The scale exerts an upward force on her equal to its reading.) (b) What is unreasonable about the result? (c) Which premise is unreasonable, or which premises are inconsistent?

A half-full recycling bin has mass 3.0 kg and is pushed up a $40.0 °$ incline with constant speed under the action of a 26-N force acting up and parallel to the incline. The incline has friction. What magnitude force must act up and parallel to the incline for the bin to move down the incline at constant velocity?

An object with mass m moves along the x-axis. Its position at any time is given by $x \left(t\right) = p t^{3} + q t^{2}$ where p and q are constants. Find the net force on this object for any time t.

Located at the origin, an electric car of mass m is at rest and in equilibrium. A time dependent force of $\overset{\rightarrow}{F} \left(t\right)$ is applied at time $t = 0$, and its components are $F_{x} \left(t\right) = p + n t$ and $F_{y} \left(t\right) = q t$ where p, q, and n are constants. Find the position $\overset{\rightarrow}{r} \left(t\right)$ and velocity $\overset{\rightarrow}{v} \left(t\right)$ as functions of time t.

A 2.0-kg object has a velocity of $4.0 \hat{i} \text{m}/\text{s}$ at $t = 0 .$ A constant resultant force of $\left(2.0 \hat{i} + 4.0 \hat{j}\right) \text{N}$ then acts on the object for 3.0 s. What is the magnitude of the object’s velocity at the end of the 3.0-s interval?

A box is dropped onto a conveyor belt moving at 3.4 m/s. If the coefficient of friction between the box and the belt is 0.27, how long will it take before the box moves without slipping?

A crate of mass 100.0 kg rests on a rough surface inclined at an angle of $37.0 °$ with the horizontal. A massless rope to which a force can be applied parallel to the surface is attached to the crate and leads to the top of the incline. In its present state, the crate is just ready to slip and start to move down the plane. The coefficient of friction is $80 \%$ of that for the static case. (a) What is the coe…

A 30.0-g ball at the end of a string is swung in a vertical circle with a radius of 25.0 cm. The tangential velocity is 200.0 cm/s. Find the tension in the string: (a) at the top of the circle, (b) at the bottom of the circle, and (c) at a distance of 12.5 cm from the center of the circle $\left(r = 12.5 \text{cm}\right) .$

A stunt cyclist rides on the interior of a cylinder 12 m in radius. The coefficient of static friction between the tires and the wall is 0.68. Find the value of the minimum speed for the cyclist to perform the stunt.

A piece of bacon starts to slide down the pan when one side of a pan is raised up 5.0 cm. If the length of the pan from pivot to the raising point is 23.5 cm, what is the coefficient of static friction between the pan and the bacon?

An airplane flies at 120.0 m/s and banks at a $30 °$ angle. If its mass is $2.50 \times 10^{3} \text{kg},$ (a) what is the magnitude of the lift force? (b) what is the radius of the turn?

A box rests on the (horizontal) back of a truck. The coefficient of static friction between the box and the surface on which it rests is 0.24. What maximum distance can the truck travel (starting from rest and moving horizontally with constant acceleration) in 3.0 s without having the box slide?

In a later chapter, you will find that the weight of a particle varies with altitude such that $w = \frac{m g r_{0} ^{2}}{r^{2}}$ where $r_{0} ^{}$ is the radius of Earth and r is the distance from Earth’s center. If the particle is fired vertically with velocity $v_{0} ^{}$ from Earth’s surface, determine its velocity as a function of position r. (Hint: use $a^{} d r = v^{} d v ,$ the rearrangement mentioned in the text.)

A car of mass 1000.0 kg is traveling along a level road at 100.0 km/h when its brakes are applied. Calculate the stopping distance if the coefficient of kinetic friction of the tires is 0.500. Neglect air resistance. (Hint: since the distance traveled is of interest rather than the time, x is the desired independent variable and not t. Use the Chain Rule to change the variable: $\frac{d v}{d t} = \frac{d v}{d x} \frac{d x}{d t} = v \frac{d v}{d x} . \left.\right)$

A skydiver is at an altitude of 1520 m. After 10.0 seconds of free fall, he opens his parachute and finds that the air resistance, $F_{\text{D}}$, is given by the formula $F_{\text{D}} = − b v ,$ where b is a constant and v is the velocity. If $b = 7.50 ,$ and the mass of the skydiver is 82.0 kg, first set up differential equations for the velocity and the position, and then find: (a) the speed of the skydiver when the parachute opens, (b) the…

A boater and motor boat are at rest on a lake. Together, they have mass 200.0 kg. If the thrust of the motor is a constant force of 40.0 N in the direction of motion, and if the resistive force of the water is numerically equivalent to 2 times the speed v of the boat, set up and solve the differential equation to find: (a) the velocity of the boat at time t; (b) the limiting velocity (the velocity…