Motion: Distance and Displacement

Key Concepts

• Distance and displacement
• Motion in a straight line
• Motion in a circular path
• Speed and average speed
• Velocity and average velocity

Introduction:  

• Distance is the length of the entire path traveled by an object from an initial to a final point. However, displacement is the shortest distance between the two points.  
• Displacement is always less than or equal to the distance covered by an object. 
• There can be many paths by which an object can move from one point to another. Each path has a different length than the other.  
• The length of the path taken by an object to reach the final point from the initial point is the distance covered by the object. Whereas the displacement is the length of the path that connects the two points directly.  
• Therefore, whichever path the object takes, the displacement remains the same, even if the distance changes.  
• Both distance and displacement are measured in the units of meters, kilometers, miles, etc.  
• However, the SI unit of measurement for both the quantities is “meter.” 

In figure 2.1 below, Alex wants to visit Martin. There are many paths that Alex can choose from to reach Martin’s place, i.e., 1, 2, 3, 4, and 5. The shortest path among them is path no. 5, which connects Alex’s and Martin’s places. Therefore, the distance between Alex and Martin is given by the length of any of the paths. 

However, the displacement is only given by the length of path no. 5. 

Motion in A Straight Line 

The simplest type of motion is motion in a straight line. For example, a vehicle moving on a perfectly straight road, a rocket moving vertically upward in a straight line, etc. Let us consider a car moving on a straight road, as shown in the figure below. 

Case 1:  

The car moves from P to S via Q and R and then returns to R. We can calculate the total distance and displacement travelled by the car in the whole journey as follows: 

Distance = PQ + QR + RS + SR 

                 = 40 + 45 + 25 + 25 

                 = 135 km 

Displacement = PR = 85 km 

Here, the distance and displacement are not equal. In fact, the distance is greater than the displacement. 

Case 2:  

Similarly, we can calculate the distance and displacement if the car further moves to point Q from R as follows: 

Distance = PQ + QR + RS + SR + RQ 

                 = 40 + 45 + 25 + 25 + 45 

                 = 180 km 

Displacement = PQ = 40 km 

Again, the distance and the displacement are not equal. 

Case 3: 

If the car does not take a turn, i.e., it goes only up to the point R, let us see what the distance and the displacement of the car will be. The situation is shown in figure 2.4 below. 

Distance = PT = 65 km 

Displacement = PT = 65 km 

Here, the distance and the displacement are equal. 

Case 4: Motion in a circular path 

Now let us calculate the distance and the displacement for a simple case of motion in a circular path of length 100 m. An object ‘A’ makes 5 complete turns around the whole path. Let us calculate the distance and the displacement of ‘A.’ 

The distance = 5 x 100 = 500 m  

The displacement = 0 m as A comes back to the same point even after making 5 turns. 

Therefore, the distance and the displacement, in this case, are not equal at all. 

From here, we can conclude that the displacement of an object can be zero sometimes. Still, its distance covered can never be zero unless the body is at rest. 

Speed 

Speed is a measure of the fastness of a motion. It can be measured in the units of km/h, m/s, miles/h, etc. However, the SI unit of speed is m/s. Mathematically, 

Speed = Distance/Time taken 

Let us calculate the speed of Daniel and Joseph, who are running a 500 m race. Daniel completed the race in 100 seconds. Whereas Joseph completed it in 110 seconds. 

Distance covered by both = 500 m 

        Time taken by Daniel = 100 s 

       Time taken by Joseph = 110 s 

                   Speed of Daniel = 500/100 = 5 m/s 

                   Speed of Joseph = 500/110 = 4.54 m/s 

Therefore, Daniel runs faster than Joseph. 

Average Speed 

In practical cases, the speed of a vehicle usually does not stay constant throughout the motion. That is when the motion is called a non-uniform motion. The data about the speed of a car throughout its journey for different time intervals is given in the table below to illustrate the same. 

In such cases, the average speed of the moving object is calculated.   

Mathematically, 

Average speed = (Total distance)/(Total time) 

Average speed is also measured in the units of km/h, m/s, miles/h, etc. However, its SI unit is also m/s. 

Example 1: Find the average speed of the rider from the data given in the figure below. 

Solution: 

Average speed = Total distance travelled/Total time taken 

                             = (800 + 900 +500 + 500 + 900)/ (5 + 8 + 2 + 9 + 21) 

                             = 3600/45 

                             = 80 m/minute 

                             = 80/60 m/s        (1 minute = 60 seconds) 

                             = 1.33 m/s 

Example 2: 

An object travels first 20 m in 5 seconds and the next 20 m in 3 seconds. Calculate its average speed. 

Solution: 

Average speed = (20 + 20)/ (5 + 3) 

                             = 40/8 

                             = 5 m/s 

Velocity 

The speed of an object moving in a specific direction is called velocity. The velocity of a body changes when there is a change in its speed, direction, or both. Velocity is expressed as speed along with its direction. For example, “the velocity of the moving truck is 50 m/s towards the west.” 

Mathematically velocity is given by, 

Velocity, v = Displacement/ Time taken 

Here, displacement is used instead of distance, unlike the formula of speed. 

The SI unit of measurement of velocity is also m/s, like that of speed. However, it can be measured in units of km/h, miles/h as well.  

If the speed or velocity is given in km/h, we can always convert it into m/s if required. This is done as follows: 

1 km/h = 1000 m/3600 s 

               = 10/36 m/s 

1 km/h = 5/18 m/s 

Example 3: 

Convert 72 km/h to m/s. 

Solution: 

1 km/h = 5/18 m/s 

72 km/h = (5/18) * 72 m/s 

                  = 20 m/s 

Example 4: 

Calculate the velocity of the car from the data shown below. The car moves from P to S via Q and R and then moves back to R. 

Solution: 

Distance travelled = 135 km 

             Time taken = 5 hours 

                       Speed = 135/5 = 27 km/h 

        Displacement = 85 km 

            Time taken = 5 hours 

                  Velocity = 85/5 = 17 km/h, east 

Example 5: 

Convert the resulting speed and velocity from example 4 to m/s. 

Solution: 

Speed = 27 km/h 

            = (5/18) * 27 m/s 

            = 7.5 m/s 

Velocity = 17 km/h 

            = (5/18) * 17 m/s 

            = 4.72 m/s 

Example 6: 

Calculate the velocity of the car from the data shown above in figure 2.8. The car moves from P to S via Q and R and then moves back to Q via R. 

Solution: 

Distance travelled = 180 km 

             Time taken = 4 hours 

                       Speed = 180/4 = 45 km/h 

        Displacement = 40 km 

             Time taken = 4 hours 

                   Velocity = 40/4 = 10 km/h, east 

Average Velocity 

Suppose an object travels in a straight line with its velocity changing uniformly. In that case, its average velocity is the arithmetic mean of its initial and final velocity. 

Mathematically, 

Average velocity = (Initial velocity + Final velocity)/2 

Example 7: 

A car is moving at a speed of 5 m/s on a straight road. It is uniformly increasing its velocity and has finally reached a velocity of 11 m/s. What is the average velocity of the car? 

Solution: 

Initial velocity = 5 m/s 

  Final velocity = 11 m/s 

Therefore, average velocity = (5 + 11)/2 

                                                      = 8 m/s 

Some physical quantities like distance, speed, and average speed do not include direction. Whereas the other physical quantities like displacement, velocity, and average velocity possess a direction. The former is a set of scalar quantities, and the latter is a set of vector quantities. Vector quantities have a magnitude as well as direction. On the other hand, scalar quantities only have a magnitude. 

Summary

• Displacement is the shortest distance between two points.
• The S I unit of distance and displacement is m.
• Distance ≥ Displacement
• The rate of change of distance is called speed.
• The rate of change of displacement is called velocity.
• A Speed with direction is called velocity.
• The S I unit of speed and velocity is m/s.
• Most motions are non-uniform, which involves changing velocity throughout the motion.
• In the case of non-uniform motion, we calculate average speed.
• 1 km/h = (5/18) m/s.