definitions of these three quantities. We then treat projectile motion and uniform circular
motion as special cases of motion in two dimensions. We also discuss the concept of
relative motion, which shows why observers in different frames of reference may measure
different positions, velocities, and accelerations for a given particle.
4.1 The Position, Velocity, and
displacement is a vector, and the displacement of the particle is
the difference between its final position and its initial position. We now define the displacement
vector for the particle of Figure 4.1 as being the difference between its
final position vector and its initial position vector:
The direction of is indicated in Figure 4.1. As we see from the figure, the magnitude of
is less than the distance traveled along the curved path followed by the particle.
We define the average velocity of a particle during the time interval !t as the displacement
of the particle divided by the time interval:
The instantaneous velocity v is defined as the limit of the average
velocity / as approaches zero:
Knowing the velocity at these points allows us to determine the average acceleration of the particle—the average
acceleration of a particle as it moves is defined as the change in the instantaneous
velocity vector v divided by the time interval t during which that change occurs:
When the average acceleration of a particle changes during different time intervals,
it is useful to define its instantaneous acceleration. The instantaneous acceleration
a is defined as the limiting value of the ratio approaches zero:
In other words, the instantaneous acceleration equals the derivative of the velocity vector
with respect to time.
4.2 Two Dimensional Motion with Constant Acceleration
Two dimensional motion during which the acceleration remains constant in both magnitude and direction. It will be useful for analyzing some common types of motion.
The position vector for a particle moving in the xy plane :
r = x iˆ + y jˆ (4.6)
· Where x, y and r change with time as a particle moves
· iˆ and jˆ remain constant
* velocity vector as a function of time
If the position vector is known, the velocity of the particle can be obtained from equations 4.3 and 4.6
Therefore, substituting from equation 2.9 , and
into equation 4.7
*Graphical Representation of Equations 4.8
is generally not along the direction of either Vi or a because the relationship between of these quantities is a vector expression. We may write in component form :
*Position Vector as a Function of time
The x and y coordinate of a particle moving with constant acceleration are
Substituting these expressions into equation 4.6 , r = x iˆ + y jˆ which give
This equation tells us that the position vector is the vector sum of the original position a displacement arising from the initial velocity of the particle and a displacement resulting from the constant acceleration of the particle.
*Graphical Representation of Equations 4.9
is generally not along the direction of Vi or a because it is vector expressions. We may write them in component form
4.3 Projectile Motion
Anyone who has observed a baseball in motion has observed projectile motion. The
ball moves in a curved path, and its motion is simple to analyze if we make two assumptions:
(1) the free-fall acceleration g is constant over the range of motion and is directed
downward,1 and (2) the effect of air resistance is negligible.2 With these assumptions,
we find that the path of a projectile, which we call its trajectory, is always a
Therefore, the initial x and y components of velocity are:
Horizontal Range and Maximum Height of a Projectile
The distance R is called the horizontal range of the projectile,
and the distance h is its maximum height.
4.4 Uniform Circular Motion
A car moving in a circular path with constant speed v. Such motion is
called uniform circular motion, and occurs in many situations. It is often surprising
to students to find that even though an object moves at a constant speed in a circular
path, it still has an acceleration.
consider the diagram of the position and velocity vectors in Figure 4.17b. In addition, the figure shows the vector representing the change in positon
The particle follows a circular path, part of which is shown by the dotted curve.
The particle is at at time and its velocity at that time is . It is at at some later time and its velocity at that time is
In order to calculate the acceleration of the particle, let us begin with the defining equation for average acceleration (Eq. 4.4):
In both Figures 4.17b and 4.17c, we can identify triangles that help us analyze the motion. the two triangles
are similar. (Two triangles are similar if the angle between any two sides is the same for both triangles and if the ratio of the lengths of these sides is the same.)
This enables us to write a relationship between the lengths of the sides for the two triangles
This equation can be solved for and the expression so obtained can be substituted into
to give the magnitude of the average acceleration over the time interval for the particle to move
As and approach each other, approaches zero, and the ratio approaches
the speed v. In addition, the average acceleration becomes the instantaneous acceleration at point
Hence in the limit the magnitude of the acceleration is
Thus, in uniform circular motion the acceleration is directed inward toward the center
of the circle and has magnitude
4.5 Tangential and Radial Acceleration
This vector can be resolved into two components, based on an origin at the center of the dashed circle: a radial component ar along the radius of the model circle, and a tangential component at perpendicular to this radius.
The total acceleration vector a can be written as the vector sum of
the component vectors:
The tangential acceleration component causes the change in the speed of the
particle. This component is parallel to the instantaneous velocity, and is given by
The radial acceleration component arises from the change in direction of the velocity
vector and is given by
4.6 Relative Velocity and Relative Acceleration
We find that observers in different frames of reference may measure different positions, velocities, and accelerations for a given particle.
As an example
Galilean coordinate transformation
Galilean Velocity Transformation
Although observers in two frames measure different velocities for the particle, they measure the same acceleration when is constant. We can verify this by taking the time derivative of Equation 4.22:
That is, the acceleration of the particle measured byan observer in one frame of reference is the same as that measured by any other observer moving with constant velocity relative to the first frame.