A. Uniform Motion - Constant Speed - no acceleration

  This is a typical graph of  the relationship between position and time for an object moving at constant speed.

d t graphBy calculating the slope m of the position-time graph we obtain the velocity of the moving object 

note Note that "m" is the slope of a straight line (of the form, y = mx + b) and it is constant.  

The slope of this line  (the average speed) can be calculated by several methods.

1. Slope Equation

slope eqn

2.   Average speed equation

aver speed eqn

 B. Uniformly Accelerated Motion - Variable Speed - Speed is not constant

d t square

This graph illustrates the relationship between the position and the time for an object whose velocity is changing with time.
We know that the velocity is changing as time goes on because the slope of this line is not constant.

note This is a  function of the form y = ax2 + bx + c.  Distance increases in a non-linear fashion.
noteTo analyze this function mathematically one would need to take the first derivative of the equation using Calculus.

At this stage we can still obtain a lot of information from this graph by using graphical analysis techniques.

1. Average velocity

To obtain the average velocity between two points in time (say t1 and t2) we can draw the secant line between these two points and calculate the slope of the secant line.

The slope of the secant line can be calculated by the usual method:

d t graph accel




This gives us the average velocity between the  time interval from  t1 to t2

Therefore we can say that: 

notevav  (we used v to simplify is the slope of the secant and it can also be calculated using the equation below: 

aver speed eqn

2. Instantaneous Velocity 

To obtain the instantaneous velocity, that is, the velocity at one instant (one point in time -- say t1), one must take the slope of the tangent line that touches the curve at that point (and that point only).

i.e. Vinst = slope of tangent

Vinst = v  or:

tangent graph accel

noteIt is also important to note that the only time where the average speed (or velocity) and the instantaneous speed (or velocity) have the same value is at the "half time" point.  Here the slope of the average secant line for the entire graph is the same as the slope of the tangent at the half time  (t1/2) point as illustrated by the animation below.   Here the half time occurs at 3s and at that point  the average velocity for the whole 6s is equal to the instantaneous velocity at 3s.  

note Note how the secant for the entire graph is the same as the tangent at the half time point.


Click the play button to view animation.  



C. Acceleration

1. Non-uniform Acceleration

Using a method similar to deriving velocity from distance-time graphs,   we can obtain values of acceleration from velocity-time graphs.

The slope of the tangent to a v-t graph at a specific point in time (at an instant) will give the instantaneous acceleration.



2. Uniform Acceleration

If the acceleration is constant the v-t graph will be a straight line.

v t graph

The slope of this line will give us the average acceleration.

 aav slope