Historical definition

Italian physicist Galileo Galilei is usually credited with being the first to measure speed by considering the distance covered and the time it takes. Galileo defined speed as the distance covered per unit of time.^[3] In equation form, that is

where ${\displaystyle v}$ is speed, ${\displaystyle d}$ is distance, and ${\displaystyle t}$ is time. A cyclist who covers 30 metres in a time of 2 seconds, for example, has a speed of 15 metres per second. Objects in motion often have variations in speed (a car might travel along a street at 50 km/h, slow to 0 km/h, and then reach 30 km/h).

Instantaneous speed

Speed at some instant, or assumed constant during a very short period of time, is called instantaneous speed. By looking at a speedometer, one can read the instantaneous speed of a car at any instant.^[3] A car travelling at 50 km/h generally goes for less than one hour at a constant speed, but if it did go at that speed for a full hour, it would travel 50 km. If the vehicle continued at that speed for half an hour, it would cover half that distance (25 km). If it continued for only one minute, it would cover about 833 m.

In mathematical terms, the instantaneous speed ${\displaystyle v}$ is defined as the magnitude of the instantaneous velocity ${\displaystyle {\boldsymbol {v}}}$, that is, the derivative of the position ${\displaystyle {\boldsymbol {r}}}$ with respect to time:^[2][4]

$${\displaystyle v=\left|{\boldsymbol {v}}\right|=\left|{\dot {\boldsymbol {r}}}\right|=\left|{\frac {d{\boldsymbol {r}}}{dt}}\right|\,.}$$

If ${\displaystyle s}$ is the length of the path (also known as the distance) travelled until time ${\displaystyle t}$, the speed equals the time derivative of ${\displaystyle s}$:^[2]

$${\displaystyle v={\frac {ds}{dt}}.}$$

In the special case where the velocity is constant (that is, constant speed in a straight line), this can be simplified to ${\displaystyle v=s/t}$. The average speed over a finite time interval is the total distance travelled divided by the time duration.

Average speed

As an example, a bowling ball's speed when first released will be above its average speed, and after decelerating because of friction, its speed when reaching the pins will be below its average speed.

Different from instantaneous speed, average speed is defined as the total distance covered divided by the time interval. For example, if a distance of 80 kilometres is driven in 1 hour, the average speed is 80 kilometres per hour. Likewise, if 320 kilometres are travelled in 4 hours, the average speed is also 80 kilometres per hour. When a distance in kilometres (km) is divided by a time in hours (h), the result is in kilometres per hour (km/h).

Average speed does not describe the speed variations that may have taken place during shorter time intervals (as it is the entire distance covered divided by the total time of travel), and so average speed is often quite different from a value of instantaneous speed.^[3] If the average speed and the time of travel are known, the distance travelled can be calculated by rearranging the definition to

$${\displaystyle d={\boldsymbol {\bar {v}}}t\,.}$$

Using this equation for an average speed of 80 kilometres per hour on a 4-hour trip, the distance covered is found to be 320 kilometres.

Expressed in graphical language, the slope of a tangent line at any point of a distance-time graph is the instantaneous speed at this point, while the slope of a chord line of the same graph is the average speed during the time interval covered by the chord. Average speed of an object is Vav = s÷t

Difference between speed and velocity

Speed denotes only how fast an object is moving, whereas velocity describes both how fast and in which direction the object is moving.^[5] If a car is said to travel at 60 km/h, its speed has been specified. However, if the car is said to move at 60 km/h to the north, its velocity has now been specified.

The big difference can be discerned when considering movement around a circle. When something moves in a circular path and returns to its starting point, its average velocity is zero, but its average speed is found by dividing the circumference of the circle by the time taken to move around the circle. This is because the average velocity is calculated by considering only the displacement between the starting and end points, whereas the average speed considers only the total distance travelled.

Tangential speed

This section is an excerpt from Tangential speed.[edit]

Angular speed and tangential speed on a disc

Tangential speed is the speed of an object undergoing circular motion, i.e., moving along a circular path.^[6] A point on the outside edge of a merry-go-round or turntable travels a greater distance in one complete rotation than a point nearer the center. Travelling a greater distance in the same time means a greater speed, and so linear speed is greater on the outer edge of a rotating object than it is closer to the axis. This speed along a circular path is known as tangential speed because the direction of motion is tangent to the circumference of the circle. For circular motion, the terms linear speed and tangential speed are used interchangeably, and both use units of m/s, km/h, and others.