## Instantaneous Velocity and Speed

3.2 Instantaneous Velocity and Speed

### Learning Objectives

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

- Explain the difference between average velocity and instantaneous velocity.
- Describe the difference between velocity and speed.
- Calculate the instantaneous velocity given the mathematical equation for the velocity.
- Calculate the speed given the instantaneous velocity.

### Speed and Velocity

Pause and Predict 3.2.1 |
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A ball, initially at the 1 m mark, moves 5 m in the positive x direction in 2.0 s. It then moves 3 m in the negative x direction in 1.5 s. Then it moves 7 m in the positive x direction in 3.0 s. What are the average speed and average velocity of the ball over the entire trip? |

(a) 6.8 m/s, 2.8 m/s |

(b) 2.3 m/s, 1.4 m/s |

(c) 2.8 m/s, 1.5 m/s |

(d) 2.3 m/s, 2.8 m/s |

(e) 6.8 m/s, 1.4 m/s |

Pause and Predict 3.2.2 |
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The graph shows the motion of a ball that moves 10 m in the +x direction in 2 s. The ball stays at that position for 2 s before moving 4 m in the -x direction in 2 s. Then the ball stops. What is the ball’s velocity at the instant of t = 5 s? |

(a) 1.6 m/s |

(b) -1.6 m/s |

(c) 2.0 m/s |

(d) -2.0 m/s |

(e) 0 m/s |

### Instantaneous Velocity

### Quick Calculus Review

**Practice!**

Practice 3.2.1 |
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This position versus time graph exhibits what kind of motion? |

(a) An object at rest. |

(b) A moving object that is speeding up. |

(c) An object moving at a constant speed. |

(d) A moving object that is slowing down. |

Practice 3.2.2 |
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This position versus time graph exhibits what kind of motion? |

(a) An object at rest. |

(b) A moving object that is speeding up. |

(c) An object moving at a constant speed. |

(d) A moving object that is slowing down. |

Practice 3.2.3 |
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This position versus time graph exhibits what kind of motion? |

(a) An object at rest. |

(b) A moving object that is speeding up. |

(c) An object moving at a constant speed. |

(d) A moving object that is slowing down. |

**Discuss!**

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

Suppose a particle moves along the x-axis. Its x-coordinate varies with time according to the expression *x*(*t*) = 8 + 3.8*t* + 2.5*t*^{2}, where *x* is in meters and *t* is in seconds.

(a) Determine the displacement Δ*x* of the particle in the time interval *t* = 9 s to *t* = 20 s.

(b) Calculate the average velocity in the time interval *t* = 9 s to *t* = 20 s.

(c) Find the instantaneous velocity of the particle at *t* = 18.4 s.