position and displacement
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 = -0.3t
2 + 7.2t + 28 y = 0.22t
2 - 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
D t, then its average velocity is:
Written as vector components:
The instantaneous velocity is the value that approaches in the limit as
D 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
D t, its average acceleration a avg during
D t is:
