## Drag Force and Terminal Speed

6.4 Drag Force and Terminal Speed

### Learning Objectives

By the end of this section, you will be able to:

- Express the drag force mathematically
- Describe applications of the drag force
- Define terminal velocity
- Determine an object’s terminal velocity given its mass

### Drag Forces

Like friction, the **drag force** always opposes the motion of an object. Unlike simple friction, the drag force is proportional to some function of the velocity of the object in that fluid.

The drag force is proportional to some function of the speed. Here, *b* is the drag coefficient and *n* is a number between 1 and 2.

When an object is moving at high velocity through air, the magnitude of the drag force is proportional to the square of the speed:

where *C* is the drag coefficient, (*rho*) is the density of the fluid, *A* is the area of the object facing the fluid, and *v* is the object’s speed.

**Discuss!**

Consider how you would answer these questions. Then bring this to class for a group discussion.

Calculate the ratio of the drag force on a passenger jet flying with a speed of 1000 km/h at an altitude of 10 km to the drag force on a prop-driven transport flying at half the speed and half the altitude of the jet. At 10 km the density of air is 0.38 kg/m^{3}, and at 5.0 km it is 0.67 kg/m^{3}. Assume that the airplanes have the same effective cross-sectional area and the same drag coefficient C.

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

You throw a baseball straight upward. The drag force is proportional to *v*^{2}.

In terms of *g*, what is the *y*-component of the ball’s acceleration when the ball’s speed is half its terminal speed and

(a) it is moving up?

(b) It is moving back down?

**Practice!**

Practice 6.4.1 |
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A baseball and a basketball, having the same mass, are dropped through air from rest such that their bottoms are initially at the same height above the ground, on the order of 10 m or more. Which one strikes the ground first? |

(a) The baseball strikes the ground first. |

(b) The basketball strikes the ground first. |

(c) Both strike the ground at the same time. |

Practice 6.4.2 |
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A ball is dropped from rest and feels air resistance as it falls. Which of the graphs best represents its acceleration as a function of time? |

Practice 6.4.3 |
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A ball is dropped from rest and feels air resistance as it falls. Which of the graphs best represents its vertical velocity component as a function of time? |

Practice 6.4.4 |
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A child on board a cruise ship drops a 30.0 gram marble into the ocean. If the resistive force proportionality constant is 0.500 kg/s, what is the terminal speed of the marble? |

(a) 0.147 m/s |

(b) 0.294 m/s |

(c) 0.588 m/s |

(d) 1.18 m/s |

(e) 2.35 m/s |