Graphs of Motion

Velocity-Time Graphs

In the previous section, the focus was on calculations covering Speed and Acceleration. In this section, graphs of Velocity against Time will be used find information about an object's motion. An example of a Velocity-Time graph is shown below :-

In order to fully explain the use of a Velocity-Time graph, each section above will be looked at separately. 

Start to A 

In this first section of the graph, The object starts at rest and then increases in Velocity to 20 ms-1. By using the graph to gather data, it is possible to calculate the Acceleration at this point :-

                a  =  ?

                v  =  20 ms-1

                u  =  0 ms-1

                t  =  5 s

                a  =  ( 20 - 0 ) / 5

                a  =  4 ms-2

Note - On a Velocity-Time graph, a diagonal line with a positive gradient shows a positive acceleration

A to B

In this section of the graph, the object travels at a constant Velocity of 20 ms-1

Note - On a Velocity-Time graph, a horizontal line shows a constant Velocity. 

B to C

In this section, the object starts at 20 ms-1 and then experiences a negative acceleration. The object's velocity decreases until it reaches a velocity of -7 ms-1. By using the graph to gather data, it is possible to calculate the negative Acceleration at this point :-

                a  =  ?

                v  =  -7 ms-1

                u  =  20 ms-1

                t  =  5 s

                a  =  ( -7 - 20 ) / 5

                a  =  -5.4 ms-2

Note - On a Velocity-Time graph, a diagonal line with a negative gradient represents a negative Acceleration.

Below X-Axis Motion

In the section B to C, the Velocity reached zero and then became negative. From an understanding of Speed, this makes no sense, but from an understanding of Velocity this is easily explained.

In this question, it was assumed that the object was initially moving in a forward direction. So when the Velocity became negative, this simply means that the object is now moving backwards. 

Note - In Physics (by convention) the following applies :-

Positive direction (+) : Upwards or to the Right

Negative direction (-) : Downwards or to the Left 

Distance / Displacement Traveled

By using the graph above it is also possible to calculate the Distance or Displacement an object has travelled. As these are Velocity-Time graphs, if the area under the graph is found, this will give the distance travelled. 

Note - This is due to distance being equal to V x T, so the area gives distance. 

The following diagram shows the graph above, colour-coded for ease of calculation :-

Note - To find area :-

Rectangle - Base x Height

Triangle - 1/2 x Base x Height

By finding the area of each section and adding them all together, the total distance can be found :-

Total Distance =  Area 1 + Area 2 + Area 3 + Area 4 

Total Distance =  (1/2x5x20) + (10x20) + (1/2x3x20) + (1/2x2x7) 

Total Distance =  287 m 

To find the Displacement, add together all sections above the axis, and subtract all sections below the axis :-

Displacement   =  (Area 1 + Area 2 + Area 3) - (Area 4) 

Displacement   =  [(1/2x5x20) + (10x20) + (1/2x3x20)] -  (1/2x2x7)

Displacement   =  273 m in the forward direction

Note - Displacement is a Vector, therefore must include a direction. 

Acceleration-Time Graphs

It is possible to use the Velocity-Time graphs above to represent visually the changing Acceleration of an object in the form of a Acceleration-Time graph. 

The diagram below shows a Velocity-Time graph with its corresponding Acceleration-Time graph :-

The graph above shows that :-

1. When the object's Velocity is increasing, it has a constant positive acceleration

2. When the object's Velocity is decreasing, it has a constant negative acceleration 

3. When the object travels at a constant Velocity, the acceleration is zero. 

Note - The values used on the Acceleration-Time graph can be found using the first equation of motion with the data gained from the Velocity-Time graph. 

Displacement- Time Graphs

Finally, Graphs relating Displacement and Time can also be generated. These Displacement-Time graphs generally have a curved shape as a constant Acceleration will give a linear increase in Velocity and therefore an increasing Displacement per second :-

Motion Case Study - Bouncing Ball

The graph below shows a Displacement-Time graph (with the origin being ground level) for a ball falling under Gravity:-

The graph below shows the corresponding Velocity-Time graph for a falling ball:-

A falling Ball experiences a constant Acceleration due to Gravity of 9.8 ms-2 downwards. This constant Acceleration causes the ball to increase its Velocity in the negative direction (as can be seen in the diagonal sections of the V-T graph). 

On impacting the ground, the ball experiences a large Force upwards which causes a large Acceleration upwards, causing the 'jumps' in the V-T graph, with the 'top' of each bounce occurs when the Velocity is zero. 

Note - The decreasing height and maximum Velocity is caused by Energy being lost at each bounce.