Lesson Explainer: Velocity Science

In this explainer, we will learn how to distinguish between the speed and the velocity at which an object moves between two points.

Recall that the speed of an object is the amount of distance traveled by the object per unit of time. For example, imagine that a car travels between an initial position and a final position, as shown in the following diagram

The speed of the car is equal to the distance it travels divided by the time it takes to travel this distance: speeddistancetime=.

Sometimes, the path that the object takes between its initial position and final position is not a straight line. The length of this path is the distance that the object travels.

This is different from the displacement that the object travels. The displacement is the change in position between the two points.

Displacement is a vector quantity; this means it has a magnitude and direction. The magnitude of displacement is the shortest distance on a straight line between two points.

We can label the distance the car travels and the displacement of its final position from its starting position on the following diagram.

The velocity of the car is equal to the displacement of its final position from its starting position divided by the time taken to complete this motion: velocitydisplacementtime=.

The unit of velocity is the metre per second, also written as m/s. However, like displacement, velocity is a vector quantity—this means it has both magnitude and direction.

This means that if we are told that the car is traveling at 10 m/s, we are told its speed. If we are told that the car is traveling in a particular direction at 10 m/s, we are told its velocity.

Definition: Velocity

The velocity of an object traveling between an initial and final position is equal to the displacement between the two positions divided by the time taken to move between them: velocitydisplacementtime=.

When an object is traveling in a straight line and its speed remains constant, the magnitude of its velocity is equal to its speed. This is illustrated in the following diagram.

As seen, the magnitude of the displacement of the final position of the car from the starting position of the car is equal to the distance between the two points. This means that the magnitude of the velocity of the car is equal to the speed of the car, provided that the speed of the car remains constant for the duration of the motion.

We will now look at an example of an object traveling in a straight line at constant speed.

Example 1: The Speed and Velocity of an Object Traveling in a Straight Line

If an object moves in a straight line at a constant speed, which of the following is correct?

  1. The speed is the magnitude of the velocity of the object.
  2. The speed becomes a vector quantity.

Answer

The speed of an object is equal to the distance it travels divided by the time it takes to travel this distance: speeddistancetime=.

The velocity of an object is equal to the displacement between its final position and its initial position divided by the time it takes to move between the two points: velocitydisplacementtime=.

The magnitude of the velocity of the object will only equal the speed of the object when the magnitude of the displacement of the points it is traveling between is equal to the distance between the points.

Between two points, the distance will only equal the magnitude of the displacement when the object travels in a straight line. This is illustrated in the following diagram.

This means that if an object moves in a straight line at a constant speed, the speed is the magnitude of the velocity of the object.

Importantly, speed is a scalar quantity; it can never be a vector quantity.

When an object travels on a curved path, the distance it travels is greater than the magnitude of the displacement. This is illustrated in the following diagram.

This means that the speed of the object is greater than the magnitude of the velocity of the object.

Example 2: The Speed and Velocity of an Object Traveling on a Curved Path

An aircraft follows the curved line shown. Which has greater magnitude, the aircraft’s speed or its velocity?

Answer

We can start by annotating the diagram to show the distance of the path taken and the displacement of the final position from the initial position. This is shown in the following diagram.

The speed of the aircraft is equal to the distance of the path taken from the initial position to the final position divided by the time taken to complete the motion: speeddistancetime=.

The velocity of the aircraft is equal to the displacement of its final position from its initial position divided by the time taken to complete the motion: velocitydisplacementtime=.

The magnitude of the distance traveled by the aircraft is greater than the magnitude of the displacement of its final position from its initial position.

This means that the speed of the aircraft is greater than the magnitude of the velocity of the aircraft.

Another interesting case to consider is when the object’s direction of motion reverses as it travels in a straight line from its initial position to its final position. This is illustrated in the following diagram, where a car is traveling on a straight, horizontal path from A to B and it reverses midway through.

We should note that the diagram shows a small amount of vertical distance traveled by the car as well as the horizontal distance traveled by the car. The car is actually supposed to have only moved horizontally. The vertical distance moved by the car is only shown in the diagram to make it easy to see the distance traveled by the car in opposite horizontal directions. For the displacement of the car, only horizontal displacement should be considered. As seen, even though the path is along a horizontal line from the initial position to the final position, the distance traveled is greater than the magnitude of the displacement between the two points.

This means that the speed of the car has a greater magnitude than the velocity of the car.

We will now work through an example where the object’s direction of motion reverses.

Example 3: The Speed and Velocity of an Object When the Direction of Motion Reverses

A car follows the red line shown. Which has the greater magnitude, the car’s speed or its velocity?

Answer

The path taken by the car shows it travels a very small vertical distance, as well as the horizontal distance it travels. The car is actually supposed to have only moved horizontally. The vertical distance moved by the car is only shown in the diagram to make it easy to see the distance traveled by the car in opposite horizontal directions. For the displacement of the car, only horizontal displacement should be considered.

The path taken by the car is along a straight, horizontal line, and the direction of motion reverses. This means that the distance traveled by the car has a greater magnitude than the displacement between the final position of the car and its initial position. This is shown in the following diagram.

As before, we should note that the diagram shows a small amount of vertical distance traveled by the car as well as the horizontal distance traveled by the car. The car is actually supposed to have only moved horizontally. The vertical distance moved by the car is only shown in the diagram to make it easy to see the distance traveled by the car in opposite horizontal directions. For the displacement of the car, only horizontal displacement should be considered. The speed of the car is equal to the distance of the path it takes divided by the time it takes to complete the motion: speeddistancetime=.

The velocity of the car is equal to the displacement of its final position from its initial position divided by the time taken to move between these two points: velocitydisplacementtime=.

The distance traveled by the car has a greater magnitude than the displacement between its final and initial positions, so the speed of the car has a greater magnitude than its velocity.

The shortest possible path between two points is a straight line. For an object traveling from its initial position to its final position, the shortest possible path the object can take is the straight line between them. As seen on the following diagram, the path with the shortest distance between two points is equal in magnitude to the displacement between the two points.

When the object travels along a straight line at constant speed, the speed of the object is the same as the magnitude of its velocity. This happens when it travels along the shortest possible path between the two points, so the speed of the object cannot be less than this.

Therefore, the speed of an object cannot be less than the magnitude of its velocity.

We will now look at an example that shows the relationship between the speed and velocity of an object.

Example 4: The Minimum Speed of an Object

Explain why the speed of an object cannot be less than the magnitude of its velocity.

Answer

The speed of an object is equal to the distance it travels from its initial position to its final position divided by the time taken to complete this motion: speeddistancetime=.

The velocity of an object is equal to the displacement of its final position from its initial position divided by the time taken to move between these two points: velocitydisplacementtime=.

Between two points, an object can take many paths; the shortest is a straight line from the initial position to the final position. This is illustrated in the following diagram.

When the object takes the shortest path, the distance it travels is equal in magnitude to the displacement of its final position from its initial position. The speed of the object along this path is also at its minimum and is equal in magnitude to the object’s velocity.

Every other path that the car can possibly take is longer, leading to a slower speed The speed of an object cannot be less than the magnitude of its velocity.

Another scenario to consider is when an object can travel on multiple paths between two points but its speed remains constant. In the following diagram, we consider two cars traveling from initial position A to final position B along two different paths.

As seen, one car takes a longer path than the other. If they are both traveling at the same speed, they will arrive at their final position B at different times.

The time taken can be calculated by dividing the distance traveled by the speed at which the object is traveling: timedistancespeed=.

So, a larger distance traveled from initial position to final position at constant speed means it takes a longer time to complete the motion.

However, the displacement of the final position from the initial position does not depend on the path taken.

The velocity of the car taking the longer path must therefore be smaller than the velocity of the car taking the shorter path.

We will now work through an example related to this.

Example 5: The Velocity of Two Objects Traveling at Constant Speed on Paths of Different Lengths with the Same Displacement

Two aircraft fly along the paths shown, each flying at the same speed. Which color arrow shows the path of the aircraft that flies between its initial and final positions at the greater velocity?

Answer

The two aircraft fly at identical speed. The aircraft on the left, taking the blue path, takes a longer path than the aircraft of the right, which takes the green path.

The time taken for an object to travel from its initial position to final position is equal to the distance of the path it takes divided by the speed it travels at: timedistancespeed=.

The aircraft on the left takes a longer path, so the time taken for it to reach its final position is longer than the time taken for the aircraft on the right.

The displacement of each aircraft’s final position from its initial position is the same. The velocity of each aircraft is equal to the displacement of its final position from its initial position divided by the time taken to complete the motion: velocitydistancetime=.

The aircraft on the left takes a longer time to complete its motion, so it has a lower velocity than the aircraft on the right. Therefore, the green color arrow shows the path of the aircraft that flies between its initial and final positions at the greater velocity.

We can summarize what we have learned in the following key points.

Key Points

  • Speed is equal to the distance traveled between initial and final positions divided by the time taken to travel this distance: speeddistancetime=.
  • Velocity is equal to the displacement of the final position from the initial position divided by the time taken to travel between these positions: velocitydisplacementtime=.
  • Velocity is a vector quantity; this means it has both direction and magnitude.
  • When an object is traveling in a straight line at a constant speed, the magnitude of its velocity is equal to its speed.
  • When an object travels along a path that is not straight, or if its direction of motion reverses, its speed is greater than the magnitude of its velocity.
  • The speed of an object cannot be less than the magnitude of its velocity.
  • If an object is traveling at a constant speed between an initial position and a final position and it takes a longer path, its velocity will decrease.

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