1.2 Speed and Velocity
Speed is a scalar quantity and does not involve direction.
Velocity is the vector related to speed; its magnitude is the speed but it also has a direction.
When an object is moving in the negative direction, its velocity is negative.
Amy has to post a letter on her way to college. The post box is 500 m east of her house and the college is 2.5 km to the west. Amy cycles at a steady speed of 10 m s−1 and takes 10 s at the post box to find the letter and post it.
Figure 1.5 below shows Amy’s journey using east as the positive direction. The distance of 2.5 km has been changed to meters so that the units are consistent.
Figure 1.5
After she leaves the post box Amy is travelling west so her velocity is negative. It is −10 m s−1.
The distances and times for the three parts of Amy’s journey are shown in Table 1.1:
Table 1.1
These can be used to draw the position–time graph using home as the origin, as in Figure 1.6.
Figure 1.6
The velocity is the rate at which the position changes.
● Velocity is represented by the gradient of the position–time graph.
Figure 1.7 is the velocity–time graph.
Figure 1.8 is the distance–time graph of Amy’s journey. It differs from the position–time graph because it shows how far she travels irrespective of her direction.
There are no negative values.
The gradient of this graph represents Amy’s speed rather than her velocity.
Figure 1.7
Figure 1.8
Note:
By drawing the graphs below each other with the same horizontal scales, you can see how they correspond to each other.
Average speed and average velocity
You can find Amy’s average speed on her way to college by using the definition:
When the distance is in meters and the time in seconds, speed is found by dividing meters by seconds and is written as m s−1. So Amy’s average speed is
Amy’s average velocity is different. Her displacement from start to finish is −2500 m. That means the college is in the negative direction.
If Amy had taken the same time to go straight from home to college at a steady speed, this steady speed would have been 6.94 m s−1.
Question 1.2.2:
What is the velocity at H, A, B and C? The speed of the marble increases after it reaches the top. What happens to the velocity?
Solution 1.2.2:
At the point A, the velocity and gradient of the position–time graph are zero. We say the marble is instantaneously at rest. The velocity at H is positive because the marble is moving in the positive direction (upwards). The velocity at B and at C is negative because the marble is moving in the negative direction (downwards).
Velocity at an instant
The position–time graph for a marble thrown straight up into the air at 5 m s−1 is curved because the velocity is continually changing.
The velocity is represented by the gradient of the position–time graph.
When a position–time graph is curved like this you can find the velocity at an instant of time by drawing a tangent as in Figure 1.9.
Figure 1.9
The velocity at P is approximately:
The velocity–time graph is shown in Figure 1.10.
Figure 1.10