In this explainer, we will learn how to find velocity as the rate of change of displacement, explaining the similarities and differences between speed and velocity.

The velocity, , of an object is related to the displacement, , of the object in a time interval, , by the formula

The formula for velocity is almost identical to the formula for speed. The only difference between them is that the formula for speed relates the change in distance, , rather than displacement, , to the change in time, .

Displacement is the straight-line distance from a point to another point. For an object that travels a straight-line path and does not change its direction, the distance traveled is the magnitude of the displacement of the object.

We can see then that, for an object that travels a straight-line path and only travels in one direction along the path, the speed of the object is the magnitude of the velocity of the object.

For an object moving in a straight-line path, the only distinction between the speed of the object and its velocity is that the velocity has a direction, from the start of the path to the end of the path, whereas the speed has no direction.

Velocity is found by dividing displacement (a vector quantity) by time (a scalar quantity). We see from this that velocity is a vector quantity. This is consistent with velocity having both a magnitude and a direction.

Just as an object can have a constant speed and an average speed, it can have a constant velocity and an average velocity. The average velocity of an object with constant velocity is equal to its constant velocity.

Let us look at an example where we determine the average velocity of an object that travels a straight-line path.

### Example 1: Determining the Velocity of an Object

What is the average westward velocity of an object that moves 11 m to the west in a time of 0.25 s?

### Answer

The question asks for the average velocity of the object. In this question, there is no way to determine the difference between the average velocity of the object and its constant velocity, so we can assume that these are equal.

The velocity, , of an object is related to the displacement, , of the object in a time interval, , by the formula

When an object travels a straight-line path, the only difference between the velocity of the object and its speed is that the velocity has a direction and the speed does not.

The question states that the velocity to be determined is westward, so it gives the direction of the velocity. The displacement of the object is stated to be to the west, in the same direction as the velocity being determined.

We need, then, only to know the distance that the object moves in this direction and the time interval in which it moves.

The magnitude of a displacement is the straight-line distance between the start point and endpoint of the motion of the object, which in this case is 11 metres.

The displacement of the object is 11 metres west, and the time interval in which the object moves is 0.25 seconds.

The westward velocity of the object is, therefore, given by

Let us now look at an example in which an object moves in a straight line but does not have a constant velocity.

### Example 2: Determining the Average Velocity of an Object

An object moves north at 12 m/s for 10 seconds and then stops and stays motionless for 10 seconds before moving north at 12 m/s for another 10 seconds. What is the objectβs average northward velocity?

### Answer

The object is initially moving, but it comes to rest and then starts to move again. The velocity of the object changes and so it cannot be constant. Rather, we are asked what the average velocity of the object is.

The average velocity, , of an object is related to the displacement, , of the object in a time interval, , by the formula

The question states that the displacement of the object is to the north and asks us what the average northward velocity is. To determine this, we need to know the total displacement northward and the time interval in which this displacement occurs.

The object travels northward at 12 m/s for 10 seconds before it comes to rest. The displacement in this time can be determined using the formula and multiplying it by ,

When the object moves for the second time, it again travels northward at 12 m/s for 10 seconds, so it must be displaced another 120 metres.

The total northward displacement of the object is given by

The time interval in which the object moves each of the 120-metre distances is 10 seconds. There are two of these time intervals. Between these time intervals is another 10-second interval in which the object is at rest.

We are determining the average northward velocity of the motion of an object from 10 seconds before the object stopped moving until 10 seconds after it started moving again. The time interval in which the object was at rest is a part of this motion.

The total time interval for the motion is, therefore, given by

Substituting the values of and obtained, we find that the average northward velocity is given by

We see that the average velocity is less than the velocity in either of the time intervals in which the object was moving. This is consistent with the fact that, for part of the total time interval for which the average velocity was determined, the object was at rest.

Let us now look at an example in which an object reverses its direction of motion.

### Example 3: Finding the Displacement of an Object with a Given Constant Speed

An object moves forward at 10 m/s for 4 s and then moves backward for 2 s at the same speed. What net forward distance does the object move from its starting position?

### Answer

The speed of the object when it moves forward is the same as when it moves backward, 10 m/s. The magnitude of the velocity both forward and backward is 10 m/s, but these velocities have opposite directions.

We must define one of these directions as positive and the opposite direction to this as negative. Let us define the forward direction as positive.

The displacement of the object in a time interval can be determined using the formula and multiplying it by ,

When the object moves forward, we find that

When the object moves backward, it has a velocity of m/s, so we find that

The total forward displacement of the object is given by

Now let us look at an example in which the average velocity of an object that reverses direction is determined.

### Example 4: Determining the Average Velocity of an Object That Reverses Direction

What is the average eastward velocity of an object that moves 6 m east and 2 m west in a total time of 0.75 s? Answer to one decimal place.

### Answer

Let us consider displacement to the east as positive.

The displacement of the object is given by

The time interval in which this displacement occurs is 0.75 seconds.

The average velocity, , of an object is related to the displacement, , of the object in a time interval, , by the formula

Substituting values of and , we find that the average eastward velocity is given by

To one decimal place, this is 5.3 m/s.

Now, let us look at an example where the velocity of an object during different parts of motion in which it reverses direction is determined.

### Example 5: Finding the Velocity of an Object for Different Parts of Its Motion

A fish swims a distance of 15 m to the left at an average speed of 2 m/s and then immediately turns around and swims half this distance to the right. The time that passes while all this occurs is 12 s, as shown in the diagram. In this question, consider displacement to the left as positive.

What is the displacement of the fish from its starting point 12 s after it starts moving?

For how long does the fish swim to the left?

What is the fishβs average velocity while it moves to the right? Answer to one decimal place.

What is the average velocity of the fish over the 12 s that the fish moves for?

### Answer

The fish is described as swimming left and then right. The diagram shows some downward motion, but this is only to make the diagram easier to read. The motion of the fish should be considered to be along a line. Traveling to the left along this line is considered positive displacement.

The total time that the fish moves for is 12 seconds. The position of the fish 12 seconds after it starts moving is its final position.

The fish is initially traveling to the left. It travels 15 m to the left. This corresponds to a displacement of 15 m.

The fish then travels to the right half of the distance that it traveled to the left. This is a distance of

The direction of the fish when it moves this distance is opposite to the positive direction, so this corresponds to a displacement of m.

The total displacement of the fish 12 seconds after it starts moving is, therefore, given by

The distance that the fish swims to the left is 15 metres. The question states that the speed of the fish when swimming to the left is 2 m/s.

The time for which the fish swims to the left can be found using the formula multiplying the formula by , and then dividing the resulting formula by :

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

The total time for which the fish swims is stated to be 12 seconds. The fish swims to the left for 7.5 seconds. The time for which the fish swims to the right is, therefore, given by

We have seen that the displacement of the fish for its motion to the right is metres. The velocity of the fish while it swims to the right is given by where is the velocity, is displacement, and is the time interval in which the fish swims to the right.

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

To one decimal place this is m/s.

The average velocity of the fish over the 12 seconds for which it moves is simply its total displacement divided by 12 seconds. The total displacement has been shown to be 7.5 metres. We see then that the average velocity is given by

The average velocity is positive as the position of the fish after 12 seconds is a point to the left of its starting position.

To summarize,

- the displacement of the fish 12 seconds after it starts moving is 7.5 m;
- the time for which the fish swims to the left is 7.5 seconds;
- the velocity of the fish while it swims to the right is m/s;
- the average velocity of the fish over the 12 seconds for which it swims is 0.625 m/s.

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

### Key Points

- The velocity, , of an object is related to the displacement, , of the object in a time interval, , by the formula
- Velocity is a vector quantity, so it has a direction and a magnitude.
- For an object traveling a straight-line path, the magnitude of the velocity of the object equals the speed of the object.
- An object that changes velocity does not have a constant velocity but does have an average velocity.
- For an object that travels along a line, one direction along the line must be defined as the positive velocity direction and the opposite direction as the negative velocity direction.
- An object that changes its direction changes its velocity, even if it maintains a constant speed.