## Average and Instantaneous Acceleration

3.3 Average and Instantaneous Acceleration

### Learning Objectives

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

- Calculate the average acceleration between two points in time.
- Calculate the instantaneous acceleration given the functional form of velocity.
- Explain the vector nature of instantaneous acceleration and velocity.
- Explain the difference between average acceleration and instantaneous acceleration.
- Find instantaneous acceleration at a specified time on a graph of velocity versus time.

### Acceleration

Average acceleration is the rate at which velocity changes:

Instantaneous acceleration is the acceleration at a specific point in time and is expressed mathematically as the derivative of the velocity function:

**Practice!**

Practice 3.3.1 |
---|

In the motion diagram, a car is driving to the right and there are equal time increments between frames. How is the car moving? |

(a) The car is slowing down. |

(b) The car is speeding up. |

(c) The car is at rest. |

(d) The car is moving at a constant speed. |

Practice 3.3.2 |
---|

In the motion diagram, a car is driving to the right and there are equal time increments between frames. How is the car moving? |

(a) The car is slowing down. |

(b) The car is speeding up. |

(c) The car is at rest. |

(d) The car is moving at a constant speed. |

Practice 3.3.3 |
---|

A race car (shown as a dot) is moving as shown in the (one dimensional) “motion diagram”. Throughout the period of time shown, what are the signs of position, , and velocity, x, vectors?v |

(a) + and + |

(b) + and – |

(c) – and + |

(d) – and – |

Practice 3.3.4 |
---|

A race car (shown as a dot) is moving as shown in the (one dimensional) “motion diagram”. Throughout the period of time shown, what is the sign of the acceleration vector, ?a |

(a) + |

(b) – |

Practice 3.3.5 |
---|

In the motion diagram, a car is moving to the right and there are equal time increments between frames. Velocity vectors are drawn to show the speed and direction of the motion. What is the direction of the car’s acceleration vector? |

(a) |

(b) |

(c) The acceleration is zero and has no direction. |

(d) It is impossible to determine the direction of the acceleration from the information given. |

Practice 3.3.6 |
---|

If an object’s acceleration vector points in the same direction as its velocity vectors, then you can conclude |

(a) The object is slowing down. |

(b) The object is moving at a constant speed. |

(c) The object is speeding up. |

(d) The object is at rest. |

Practice 3.3.7 |
---|

If an object’s acceleration vector points in the opposite direction of the velocity vectors, then you can conclude |

(a) The object is slowing down. |

(b) The object is moving at a constant speed. |

(c) The object is speeding up. |

(d) The object is at rest. |

Practice 3.3.8 |
---|

The motion of two cars, A and B, is described with these position versus time graphs. From these graphs, what can you conclude about the motion of the cars? |

(a) Car A is accelerating more than Car B. |

(b) Car B is accelerating more than Car A. |

(c) Car A is driving at a faster constant speed than Car B. |

(d) Car B is driving at a faster constant speed than Car A. |

Practice 3.3.9 |
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This velocity versus time graph exhibits what kind of motion? |

(a) An object at rest. |

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

(c) An object at rest. |

(d) A moving object that is speeding up at a constant rate. |

Practice 3.3.10 |
---|

This velocity versus time graph exhibits what kind of motion? |

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

(b) An object at rest. |

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

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

**Discuss!**

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

Estimate your acceleration if you run into a brick wall.

A car’s velocity as a function of time is given by * v*(

*t*) = (3 m/s) + (0.1 m/s

^{3})

*t*

^{2}.

(a) Calculate the average acceleration for the time interval *t* = 0 to *t* = 5.00 s.

(b) Calculate the instantaneous acceleration for *t* = 0 s and *t* = 5.00 s.

(c) Draw ** v** vs.

*t*and

**vs.**

*a**t*graphs for the car’s motion between

*t*= 0 and

*t*= 5.00 s.