Motion in Two and Three Dimensions

4-2 Position and Displacement

The position vector is typically used to indicate the location of a particle. The position vector is a vector that extends from a reference point (usually the origin) to the particle.

The position vector for a particle is the vector sum of its vector components.

In unit vector notation can be written as:

The coefficients x, y, and z are the scalar components.

The coefficients x, y, and z give the particle’s location along the coordinate axes and relative to the origin.

The figure shows a particle with position vector:

As a particle moves, its position vector changes in such a way that the position vector always extends to the particle from the reference point (origin).

If the position vector changes from during a certain time interval, then the particle’s displacement during that time interval is:

We can rewrite this displacement as:

Sample Problem 4-1

Sample Problem 4-1

In figure 4-2 the position vector for a particle is initially

and then later it is

What is the particle’s displacement from ?

Sample Problem 4-2

A rabbit runs across a parking lot on which a set of coordinate axes has been drawn.The coordinates of the rabbit’s position as functions of time t are given by:x = -0.3t2 + 7.2t + 28

y = 0.22t2 - 9.1t + 30

At t = 15 seconds, what is the rabbit’s position vector in unit-vector notation and as a magnitude and an angle?

Graph the rabbit’s path for t = 0 to t = 25 s.

Average Velocity and Instantaneous Velocity

If a particle moves through a displacement in time interval t, then its average

velocity is:

Written as vector components:

The instantaneous velocity is the value that approaches in the limit as t shrinks to

0.

The direction of the instantaneous velocity of a particle is always tangent to the particle’s path at the particle’s position.

The velocity of a particle along with the scalar components of

Sample Problem 4-3

For the rabbit in sample problem 4-2, find the velocity at time t = 15 s, in unit vector notation and as a magnitude and an angle.

Average Acceleration and Instantaneous

Acceleration

When a particle’s velocity changes from to

in a time interval t, its average acceleration a avg during t is: