Graphical Analysis of Linear Motion

#### 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.

The slope m of the position-time graph gives the velocity of the object even when the relation between the position (d) and the time (t) is not a straight line..

We note that "m", the slope of the line (of the form, y = mx + b) is constant and can be calculated by several methods.

1. **Slope = rise/run**

2.

#### B. Uniformly Accelerated Motion - Variable Speed

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 and the function (of the form y = ax^{2} + bx + c) is an increasing function.

To analyze this function properly one would need to take the first derivative of the function using Calculus.

It is sufficient to say, however that we can still obtain some useful information by relying on graphical analysis techniques.

**1. Average velocity**

To obtain the average velocity between two points in time (say t_{1} and t_{2}) we can draw the secant line between these two points and calculate the slope of the secant line.

The slope of the secant line (the line that cuts the curve at the two intersecting points (d_{1},t_{1} and d_{2}, t_{2}) can be calculated by the usual slope method:

slope (of secant line) = Delta d / Delta t

This gives us the average velocity between the time interval from t_{1} -to- t_{2}

**Therefore: vav = v = slope ** or,

**2. Instantaneous Velocity**

To obtain the instantaneous velocity, that is, the velocity at one instant (one point in time -- say at exactly t_{1}), one must take the slope of the tangent line that just * touches* the curve at that point.

i.e. **V _{inst }= slope of tangent**

V_{inst} = v or, using the slope equation we calulte the slope of the tangent (the red line):

**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 one point in time will give **instantaneous acceleration [a _{inst}] in the case where there is non uniform (changing) acceleration**.

**2. Uniform Acceleration**

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

The slope of this line will give us the acceleration.