## Vector Acceleration and Speed

### Position Vector

Imagine a piece of furniture moving on a random path with an O origin.

If we place a Cartesian plane situated on this origin, then we can locate the furniture in this trajectory by means of a vector.

The vector it is called displacement vector and has modulus, direction and direction.

*= P-O*

### Vector Speed

Vector Average Speed: Consider a moving along the trajectory of the chart above, occupying positions and in moments and respectively.

Knowing that the average velocity is equal to the time-shift vector quotient:

Note:

The average velocity vector has the same direction and direction as the displacement vector because it is obtained when we multiply a positive number.

by the vector .

Instantaneous Speed Vector: Analogous to instantaneous scalar speed, when the time interval tends to zero (), the calculated speed will be the instantaneous speed.

So:

### Vector Acceleration

Vector Average Acceleration: Considering a piece of furniture that travels any trajectory with speed in an instant and speed at a later time , its average acceleration will be given by:

Note:

As for the velocity vector, the acceleration vector will have the same direction and direction as the velocity vector, as it is the result of the product of this vector () by a positive scalar number, .

Instant Acceleration Vector: Instant vector acceleration will be given when the time interval tends to zero ().

Knowing these concepts, we can define the functions of velocity as a function of time, displacement as a function of time, and the Torricelli equation for vector notation:

For example:

A body moves with speed , and constant acceleration , as described below:

(a) What is the velocity vector after 10 seconds? (b) What is the position of the furniture at this moment?

*(a) To calculate vector velocity as a function of time, we need to decompose the initial velocity and acceleration vectors into their projections into x and y:*

*So we can divide the motion into vertical ( y) and horizontal (x):*

*In x:*

*In y: *

*From these values we can calculate the velocity vector:*

*(B)**Knowing the velocity vector, we can calculate the position vector by the Torricelli equation, or by the hourly displacement function, both in the form of vectors:*

By Torricelli:

in the same direction and direction as the acceleration and velocity vectors.

By Time Position Function:

in the same direction and direction as the acceleration and velocity vectors.