Physics in Motion

# Speed and Velocity in One Dimension

In the previous sections on Position, Distance, and Displacement, we explored how coordinate systems are used to describe the position of an object in space at a given time, and were introduced to the position function. Now, we would like to go beyond that and begin describing the behaviour of that object as it moves through space and time.

### Speed

Naturally, as an object moves through space, we would like to know what distance an object would move over a given period of time. In other words, we would like to know the rate at which its position per unit of time. This is known as the speed of the object, and allows us to describe how fast an object is moving.

For example, an object that is moving at a speed of

$$10 \ \frac{m}{s}$$

will move 10 metres every second.

### Average and Constant Speed

If we take the total distance travelled by an object and divide it by the time that it took for the object to travel that distance, we would get the average speed of that object:

$$average \ speed = \frac{p_2 - p_1}{t_2 - t_1} = \frac{ \Delta p }{ \Delta t }$$

With average speed, we only know the total distance travelled, and the time that it took to travel that distance. We do not know any information about the speed of the object at any specific point in time. An object may have constant speed throughout that time interval, or variable speed, and it is possible for two different objects to have the same average speed, but different speeds throughout the periods of time that they were travelling.

On the left, we can see two particles travelling to the left. Initially, they travel at the same speed. At around a distance of 1 from the origin, the grey particle beings moving faster, before slowing down to match the speed of the black particle. They both reach a distance of 4 units from the origin at the same time. Although their speeds are different between units 0 - 4, the calculation of their average speeds returns the same result.

### Instantaneous Speed

In order to capture the full picture of how an object moves from its starting position to its final position within a period of time, we consider what we call instantaneous speed, which is a snapshot of what an object's speed is at any specific point in time.

In the case of constant speed, the object's instantaneous speed will be the same at every point in time. In the case of variable speed, the object's instantaneous speed may be different at different points in time, as we see in the visualization above.

Through instantaneous speeds, we are able to capture a more complete picture of an object's displacement in time.

### Velocity

When we refer only to the speed of an object, we do not include any information on the direction in which the object is travelling. When we know both the speed and direction of an object, we call this value the velocity of the object.

Unlike speed - which will always be positive - values of velocity can be either positive or negative. In one dimension, this tells us the direction that the object is travelling in, as there are only two directions in which the object may move in. We consider positive velocities one direction, and negative velocities the opposite direction. If we ignore the direction of velocity (i.e. take the magnitude of velocity), we get the speed.

For example, on the left we see one object moving to the right, which we consider the positive direction, and another object moving to the left, which we consider the negative direction.

### Average and Constant Velocity

Average velocity is the total dispacement of an object divided by the interval of time that it took for the object to be displaced:

$$v_{average} = \frac{p_2 - p_1}{t_2 - t_1} = \frac{\Delta p}{\Delta t}$$

Similar to average speed, it is possible for the velocity of an object to be either constant, or variable through the time interval that is used to calculate its average velocity.

When we graph the position function of an object moving with constant velocity, it will appear as a straight line, which shows that the change in position is the same for each change in unit of time. Notice that calculating slope of the graph is the same as calculating the average velocity:

When we graph the position function of an object moving with variable velocity, it will appear as a curve on the graph. However, if the starting and ending points are the same as another object moving with constant velocity, then their average velocities will be the same:

### Instantaneous Velocity

Similar to instantaneous speed, instantaneous velocity is a snapshot of what an object's velocity is at a specific point in time.

More formally, the velocity of an object is defined as the derivative of its position function:

$$v(t) = \lim_{\Delta t \to 0} \frac{\Delta p}{\Delta t} = \frac{d p(t)}{dt}$$

This describes how fast something moves, and in which direction it is moving at any point in time. On a graph, the instantaneous velocity of an object is equal to the tangent of the position function at a point in time.