Velocity–Time graphs

Introduction

In busy costal areas, shipping lanes are monitored closely to prevent accidents. The paths of ships are tracked carefully to chart courses avoiding other traffic.

The speed of a ship and the direction in which it is travelling are the key pieces of information required for shipping lanes to operate safely. Global positioning systems specify the positions of ships, and navigation aids determine their velocity. Information about the ship's velocity can be presented in graphical form for review when required.

This unit will consider the information that can be determined from a velocity versus time graph.

Velocity–time graphs

Grab the ball in the simulation of Fig.1 and drag it to the top of its range of motion. Release the ball and observe its motion and the velocity–time graph. Then answer the following questions.

Click on the figure below to interact with the model.

Figure 1. Accelerating ball.

A bouncing ball

Grab the ball in the simulation of Fig.2 and drag it to the top of its range of motion. Release the ball and observe its motion and the velocity–time graph. At first the ball falls, but then it bounces. Use the pause button to investigate how the ball's velocity changes as it bounces.

Click on the figure below to interact with the model.

Figure 2. Bouncing ball.

At the top of each bounce the speed is zero for an instant, because the ball changes direction. At the points where the speed is zero, the gradient of a distance–time graph would be zero. A tangent to the distance–time curve at these points would be a horizontal line.

From the velocity–time graph in Fig.2 we can also determine times when the ball is stationary, speeding up, or slowing down. Additionally we can specify occasions when it is rising or falling. All this information is conveyed in the velocity–time graph.

Complete the following statements to summarize the behaviour of the ball in Fig.2.

As the ball falls towards the floor its speed . As the ball rises towards it highest point its speed .

Just before it hits the floor its speed is maximum and it is moving . Immediately after leaving the floor, at the start of an upwards journey, its velocity is maximum but is in the direction.

Since the graph in Fig.2 is plotting the ball's velocity, the direction in which it is travelling is important.

The motion of the bouncing ball can be summarized as follows:

When the ball is released at the top of its range of motion, it falls. Its velocity starts at zero and becomes more and more negative as it speeds up on the way down (section OA on Fig.3).

Its velocity as it falls goes from 0 ms⁻¹ to its maximum negative value just as it hits the floor (at point A on Fig.3).

Contact with the floor is represented by section AB on Fig.3.

After leaving the floor, it sets off upwards and once again it slows as it rises (section BC on Fig.3).

As the ball slows down when rising, its velocity gets less and less positive until, at the highest point, it is zero and the ball is stopped momentarily (at point C on Fig.3).

Figure 3.	Graph for one bounce.

Clearly the bouncing ball is one case where the velocity–time graph conveys a lot of information about speeding up, slowing down, rising, or falling.

Determining distance travelled

Set the initial speed of the car in Fig.4 to 20 ms⁻¹ and the reaction time to 1 second. Press the button to stop the car, and note the shape of the graph as the car slows to a halt. Initially the car travels at a steady speed for a short time while the driver's braking foot reacts to the signal from his brain to stop. The length of the horizontal part of the line in the graph indicates this 'reaction time'.

Figure 4.	Stopping distances.

Altering the initial speed and reaction time alters the thinking distance. If the initial speed is v and the thinking time t, then the thinking distance d is given by

Note also that the thinking distance is equal to the area under the horizontal part of the line on the graph.

Set the initial speed of the car set to 30 ms⁻¹ and the reaction time to 1.5 seconds.

The thinking distance is again given by the area under the horizontal part of the line on the graph. Additionally, the distance travelled while decelerating is given by the area under the sloped part of the line. This area is a right angled triangle and its size can be calculated from its length and height.

The total distance required to stop the car is the thinking distance plus the braking distance. This is the total area under the velocity–time graph.

Determining acceleration

The

acceleration

An object's acceleration is its rate of change of velocity.acceleration of a moving object is defined as

In the simulation in Fig.5, the braking force used to bring the car to rest can be altered.

Set the car's initial speed in Fig.5 to 30 ms⁻¹, the reaction time to 1 second, and the braking force to 4 kN.

Figure 5.	Variable deceleration.

Set the car's initial speed in Fig.5 to 30 ms⁻¹, the reaction time to 1 second, and the braking force to 6 kN.

Once again, the area under the graph in this example indicates the distance travelled while decelerating. Additionally, the steepness of the line indicates how quickly the velocity is changing.

The actual value for the deceleration can be found by calculating the gradient of the velocity–time graph. The decelerating car produced a velocity–time graph that was a straight line with a negative gradient.

Summary

From velocity–time graphs it is possible to identify when objects are speeding up, slowing down, or travelling at a constant speed.

When an object's velocity changes from a positive value to a negative one, the direction in which the object is moving has changed.

The area under the line in a velocity–time graph gives the distance travelled.

The gradient of the line in a velocity–time graph gives the acceleration. Large accelerations are represented by steep gradients.

Exercises

Figure 6.

Figure 7.

Figure 8.	Car A.

Figure 9.	Car B.

Figure 10.

	Instantaneous speed / ms⁻¹	Average speed / ms⁻¹
A	16.0	16.0
B	16.0	10.7
C	12.0	6.0
D	8.0	24.0
E	12.0	8.0