Kinematics is the branch of mechanics concerned with
describing a motion.
1. Position, distance, and displacement
Figure 1-1. displacement and distance
Position
is a location of an object relative to some reference point, often the origin
(or a zero position) of a coordinate system. When an object moves from one
position(x1) to another(x2), the distance is the length of the path the
object follows and the displacement is described as the length of the
straight line from x1 to x2. The displacement is also defined as a change in
position and represented by x or r.
Let's
assume that x1 has a point (2, 3, 5) and x2 has another position (5, 9, 4) in 3
dimensional space and we want to find displacement from x1 to x2. The
magnitude of displacement can be described as follows;
2. Average velocity and average speed
Figure. 2-1
The blue
curve of Figure 2-1 is somewhat abstract and quite unlike what you would see,
but there is a lot of information inside the graph. It is also describing how
fast the object is moving. Average velocity, v, is the ratio of the
displacement that occurs during a particular time interval to that interval;
(2-1)
The average velocity is actually the slope the red line which is from the
starting point and the ending point. The velocity is a vector quantity just like
a displacement. So the velocity has a magnitude and direction too.
Average
speed, s, is defined by the total distance divided by the tame taken.
Average speed is also describing how fast an object is. While the average
velocity is related to the displacement of the object, the average speed is
associated with the distance.
(2-2)
3. Instantaneous velocity and speed
Instantaneous velocity
and speed describe how an object is fast at a given
instant. The instantaneous velocity can be obtained by shrinking the time
interval, 
, to zero. As
dwindles, the average velocity approaches a
limiting value, which is the velocity at that instant;
(2-3)
The instantaneous velocity in calculus is the rate at which an
object's position x is changing with time at a given instant. Geometrically
speaking, it is a tangent vector to the position-time curve at a given instant.
Like an average speed, the instantaneous speed is a scalar quantity.
4. Acceleration
When an
object is moving and changing its velocity, the object is said to experience
acceleration. There are also two types of acceleration, which are average
acceleration and instantaneous acceleration. Average acceleration is
described as;
(4-1)
The instantaneous acceleration is the derivative of the velocity. In
other words, the acceleration of an object at any instant is the rate at which
its velocity is changing at that instant.
(4-2)
Accelerations are sometimes expressed in terms of g
(gravitational acceleration) which is 9.81 m/s/s.
5. Constant acceleration
When
the acceleration is constant, the average acceleration and instantaneous
acceleration are equal and we can modify the equation;
(5-1)
If we modify the equation 5-2 and write the velocity, v, in terms of the
initial velocity (v0), acceleration (a), and the time taken (t);
(5-2)
In the same way, we can modify the equation 2-1 as follows;
(5-3)
and change the equation 5-3 and describe x (displacement) in terms of initial
position (x0), velocity (v), and time taken (t);
(5-4)
If the plot of v against t shows a straight line, the average velocity
can be described in terms of the initial velocity (v0) and final velocity (v1).
(5-5)
Table 5-1. Equations for motion with constant acceleration
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6. Free Fall
An
important example of straight-line motion with constant acceleration is that of
an object rising or falling freely near the Earth's surface. The constant
acceleration equations describe this motion, but we make two changes in
notation. You replace a with g (gravitational acceleration) and x with y
(vertical displacement).
Table 6-1. Equations for free fall
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