# 6.4 Drag force and terminal speed  (Page 5/12)

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## Significance

In the both of the previous examples, we found “limiting” values. The terminal velocity is the same as the limiting velocity, which is the velocity of the falling object after a (relatively) long time has passed. Similarly, the limiting distance of the boat is the distance the boat will travel after a long amount of time has passed. Due to the properties of exponential decay, the time involved to reach either of these values is actually not too long (certainly not an infinite amount of time!) but they are quickly found by taking the limit to infinity.

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

0.27 kg/m

## Summary

• Drag forces acting on an object moving in a fluid oppose the motion. For larger objects (such as a baseball) moving at a velocity in air, the drag force is determined using the drag coefficient (typical values are given in [link] ), the area of the object facing the fluid, and the fluid density.
• For small objects (such as a bacterium) moving in a denser medium (such as water), the drag force is given by Stokes’ law.

## Key equations

 Magnitude of static friction ${f}_{\text{s}}\le {\mu }_{\text{s}}N$ Magnitude of kinetic friction ${f}_{k}={\mu }_{k}N$ Centripetal force ${F}_{\text{c}}=m\frac{{v}^{2}}{r}\phantom{\rule{0.5em}{0ex}}\text{or}\phantom{\rule{0.5em}{0ex}}{F}_{\text{c}}=mr{\omega }^{2}$ Ideal angle of a banked curve $\text{tan}\phantom{\rule{0.2em}{0ex}}\theta =\frac{{v}^{2}}{rg}$ Drag force ${F}_{D}=\frac{1}{2}C\rho A{v}^{2}$ Stokes’ law ${F}_{\text{s}}=6\pi r\eta v$

## Conceptual questions

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

The pros of wearing body suits include: (1) the body suit reduces the drag force on the swimmer and the athlete can move more easily; (2) the tightness of the suit reduces the surface area of the athlete, and even though this is a small amount, it can make a difference in performance time. The cons of wearing body suits are: (1) The tightness of the suits can induce cramping and breathing problems. (2) Heat will be retained and thus the athlete could overheat during a long period of use.

Two expressions were used for the drag force experienced by a moving object in a liquid. One depended upon the speed, while the other was proportional to the square of the speed. In which types of motion would each of these expressions be more applicable than the other one?

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 oil is less dense than the water and so rises to the top when a light rain falls and collects on the road. This creates a dangerous situation in which friction is greatly lowered, and so a car can lose control. In a heavy rain, the oil is dispersed and does not affect the motion of cars as much.

Why can a squirrel jump from a tree branch to the ground and run away undamaged, while a human could break a bone in such a fall?

## Problems

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 pike (headfirst) position with a surface area of $0.140\phantom{\rule{0.2em}{0ex}}{\text{m}}^{2}$ .

115 m/s or 414 km/h

A machine at a post office sends packages out a chute and down a ramp to be loaded into delivery vehicles. (a) Calculate the acceleration of a box heading down a 17.4° slope, assuming the coefficient of friction for a parcel on waxed wood is 0.100. (b) Find the angle of the slope down which this box could move at a constant velocity. You can neglect air resistance in both parts.
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Dimension is a quantity which give us direction of any vector unit like velocity
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