In this explainer, we will learn how to find the speed as a rate of change of distance that objects move during a time interval.

When an object travels some distance, it travels that distance in some time interval. The speed of the object depends on the distance it moves and the length of the time interval.

The distance that an object moves is the length of the path that connects the points at which the motion of the object starts and ends.

Referring to the motion of an object having points where it starts and where it ends does not necessarily mean that the object is at rest at either of these points.

For example, when a 100-metre-race is run, runners travel a distance from the start line to the finish line of the racetrack. A runner who runs the race in a straight line runs a distance of 100 metres. A runner is at rest at the start line of the racetrack when the race starts, but they are not at rest when they reach the finish line; they are likely to be running very fast over the finish line.

The speed of an object is represented by the symbol .

The speed of an object that travels a distance is related to the distance and the time that the object moves for by the formula where is the speed of the object, is the distance the object travels, and is the time for which the object travels.

The formula for the speed of an object can be, and often is, written with just and instead of and .

For distances in metres and times in seconds, the unit of speed is metres per second (m/s).

Let us look at an example of determining the speed of an object.

### Example 1: Finding the Speed of an Object given the Distance It Traveled and the Time of Travel

What is the speed of an object that travels a distance of 300 metres in a time of 25 seconds?

### Answer

The speed of the object can be determined using the formula where is the speed of the object, is the distance the object travels, and is the time for which the object travels.

Substituting the values of distance and time given, we find that

For an object that travels at a known speed, the distance traveled in a known time can be determined.

Let us look at an example in which a distance traveled by an object with a known speed is determined.

### Example 2: Calculating the Distance Traveled by an Object given Its Speed and Time of Travel

What distance does an object with a speed of 15 m/s travel in a time of 4 seconds?

### Answer

The speed of the object, , is related to the distance it travels, , and the length of the time interval for which it travels, , by the formula

This formula can be expressed with as the subject. This can be achieved by multiplying the formula by :

Substituting the values of distance and speed given, we find that

For an object that travels at a known speed, the time taken to travel a known distance can be determined.

Let us look at an example in which the time interval length for an object with a known speed is determined.

### Example 3: Calculating the Time for an Object to Travel a Given Distance at a Given Speed

How much time is taken for an object with a speed of 80 m/s to travel a distance of 300 m?

### Answer

The speed of the object, , is related to the distance it travels, , and the length of the time interval for which it travels, , by the formula

This formula can be expressed with as the subject. This can be achieved by multiplying the formula by and then dividing the resulting formula by :

Substituting the values of distance and speed given, we find that

The following figure shows the distance traveled, , and the time interval length, , of both a blue object that moves equal distances in equal times and a green object that moves unequal distances in equal times.

For the green object, the speed changes throughout its motion. The average speed of the object is found from the total distance moved and the total time taken to move that distance.

If the object travels equal distances in equal times, it has a constant speed, and this is equal to the average speed of the object. The constant speed of the object is found from the distance moved in any part of the motion of the object and the time taken to move that distance.

We notice that, in the example, both objects travel 5 metres in a time of 5 seconds. So, over a 5-second time interval that starts when the objects start to move, the average speed of the green object equals the average speed of the blue object.

For a 3-second time interval that starts when both objects start to move, again the average speeds of the green and blue objects are equal.

For a 2-second time interval that starts when both objects start to move, the green object travels a distance of 1.5 m while the blue object travels a distance of 2 m. In this time interval, the average speeds of the objects are not equal.

The formula for average speed must be expressed as rather than as

In many examples, the speed of an object could be either its constant speed or its average speed, and the example will not specify whether a speed is a constant or an average speed.

Let us now look at an example involving constant speed.

### Example 4: Determining a Distance Traveled by an Object with Constant Speed

A car is driven along a road at a constant speed. The distances that the car has moved from its starting position at different times are shown in the diagram. Find the distance to the nearest metre.

### Answer

In this example, the speed is stated to be the constant speed of the car.

The speed of the car can be determined using the formula where is the speed of the car, is the distance it travels, and is the time for which it travels.

Substituting the value of distance moved in the first 3 seconds, we find that

The car moves at this speed to travel a distance in a time interval , where is given by

The distance moved by the car in this time is given by

Substituting known values, we find that

To the nearest metre, this is 21 metres.

Let us now look at an example involving average speed.

### Example 5: Calculating the Average Speed of a Moving Object

A cat walks in a garden. The cat walks north a distance of 3 m in a time of 12 s. The cat then stops for 18 s before walking another 6 m north in a time of 15 s. What is the catβs average speed?

### Answer

In this example, the speed is stated to be the average speed of the cat.

The average speed of the cat can be determined using the formula where is the average speed of the cat, is the total distance it travels, and is the total time for which it travels.

The cat walks a total distance north given by

The total time interval forwhich the cat moves this distance is given by

Substituting the values of distance and time obtained, we find that

The distance that an object moves when its speed is determined need not be a straight-line path. Between the starting point and the ending point of the motion of an object, it can change direction.

Let us look at an example in which an object changes direction.

### Example 6: Determining the Speed of an Object That Travels between Three Points

In the diagram shown, the distance is 120 m and the distance is 280 m. At an average speed of 8 m/s, how much time is taken to move from to and then to by traveling along the two lines shown in the diagram?

### Answer

In this example, the speed is stated to be the average speed of the object that travels from to and then from to .

The speed of the object, , is related to the distance it travels, , and the length of the time interval for which it travels, , by the formula

This formula can be expressed with as the subject. This can be achieved by multiplying the formula by and then dividing the resulting formula by :

The distance traveled is the sum of the distance and the distance , so it is given by

Substituting the values of distance and speed, we find that

The points that the speed of an object is measured between can be along a curved path, such as the between points and in the following figure.

Let us look at an example in which an object travels along a curved path.

### Example 7: Calculating Speed given Distance and Time

The distance between position and position shown in the diagram is 2βββ500 m. What is the average speed of a car that travels from to in 200 seconds?

### Answer

In this example, the speed is stated to be the average speed of the object that travels from to .

The speed of the object can be determined using the formula where is the average speed of the object, is the distance the object travels, and is the time for which the object travels.

Substituting the values of distance and time given, we find that

Let us now summarize what has been learned in these examples.

### Key Points

- The speed of an object that travels a distance is related to the distance and the time that the object moves for by the formula where is the speed of the object, is the distance the object travels, and is the time for which the object travels.
- If an object moves equal distances in equal time, it has constant speed.
- If an object moves unequal distances in equal time, it has an average speed equal to the total distance moved divided by the total time taken to move that distance.
- The direction that an object moves in does not affect its speed.