## Acceleration Vector

4.2 Acceleration Vector

### Learning Objectives

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

- Calculate the acceleration vector given the velocity function in unit vector notation.
- Describe the motion of a particle with a constant acceleration in three dimensions.
- Use the one-dimensional motion equations along perpendicular axes to solve a problem in two or three dimensions with a constant acceleration.
- Express the acceleration in unit vector notation.

### Acceleration Vector in Three Dimensions

This acceleration vector is the **instantaneous acceleration** and it can be obtained from the derivative with respect to time of the velocity function:

The acceleration in terms of components is

Also, since the velocity is the derivative of the position function, we can write the acceleration in terms of the second derivative of the position function:

**Practice!**

Practice 4.2.1 |
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Consider a two-dimensional (2D) acceleration vector in the x–y plane, with magnitude a and angle measured counterclockwise from the positive x-axis. Which of these is a correct trigonometric relationship between a, , a_{x} and a_{y}? |

(a) |

(b) |

(c) |

Practice 4.2.2 |
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As an object moves in the x–y plane, which statement is true about the object’s average acceleration? |

(a) The average acceleration for some period of time equals the change in the object’s speed during that period divided by the change in time during that period. |

(b) The average acceleration for some period of time equals the change in the object’s velocity during that period divided by the change in time during that period. |

(c) The average acceleration is equal to the object’s velocity divided by the time. |

(d) The average acceleration is for some period is equal to the change in time during that period divided by the change in the object’s velocity during that period. |

**Discuss!**

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

A dog running in an open field has a velocity of

at *t*_{1} = 10.0 s. For the time interval from *t*_{1} = 10.0 s to *t*_{2} = 20.0 s, the average acceleration of the dog has magnitude 0.45 m/s^{2} and direction 31.0° measured from the +*x*-axis.

At t_{2} = 20.0 s,

(a) what are the *x* and *y* components of the dog’s velocity?

(b) What are the magnitude and direction of the dog’s velocity?

(c) Sketch the velocity vectors at *t*_{1} and *t*_{2}. How do these two vectors differ?