Displacement is always lesser than distance: Know how

• Eastern Eye Staff
• 25 November, 2020

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By: Eastern Eye Staff

The total length of the path traversed by a body is called the distance travelled by it. The direction in the distance is not always straight; it can differ accordingly. It is a scalar quantity. It is mainly represented by the letter A.

On the other hand, the shortest distance from the start to the final position of the body is called the magnitude of displacement. It is towards the direction from the initial position to the final position. It is a vector quantity and generally represented by Symbol S with a vector sign on top.

Let’s learn more about the two with the formulas of Displacement and Distance.

Formulas of Distance and Displacement

The formulas are as follows:

Distance = Speed × Time.

Displacement = Velocity x Time.

Units of Distance and Displacement

Distance-The S.I. unit of distance is metre(m), and the C.G.S. the unit is cm.

Displacement -The S.I. unit of displacement is metre(m), and the C.G.S. unit is centimetre(cm).

For example:

• Consider a car moving from Delhi to Lucknow along the highway and then another car again coming back from Lucknow to Delhi along the same highway. 
• Then the total length of the path covered by the car while moving Delhi to Lucknow and Lucknow to Delhi is called the distance moved by car. 
• While the car moves from Delhi to Lucknow, then the object’s position changes.
• This difference in the position of an object while moving is called displacement.

Difference between Displacement and Distance

• The distance can be the length of the path covered by the object in a particular period. At the same time, displacement is the distance covered by the object in a specific direction at a particular time (i.e., it is the shortest distance between the final and initial positions).
• Distance only has the magnitude, so it is a scalar quantity, whereas displacement has magnitude and direction; hence, it is a vector quantity.
• It is usually dependent on the way the object follows, whereas the displacement will not depend on the path followed by the object.
• Distance is always more than zero, i.e., positive, while displacement can be positive or negative depending on direction.
• The distance is always more than or equal to the magnitude of displacement, whereas the displacement magnitude can either be less than or equal to the distance, but it can never be greater than the distance.
• The distance can never be zero in any case, even if the displacement is zero, while displacement is always zero if the distance is equal to zero.
• Distance doesn’t depend upon time, whereas displacement decreases with time.

How is the magnitude of the displacement always lesser than that of distance?

Let’s take an example, consider a body moving from A to B along the curved path. The distance covered by the object is equal to the length of the curved path AB, but the object’s displacement is along the straight-line AB shown by the dotted arrow. The displacement will always be lesser than the distance covered by the body.

If the motion of the body is along a fixed direction, the magnitude of displacement is equal to that of distance. However, if the motion is along a curve or if the direction of motion changes, the magnitude of the displacement is less than that of distance. The magnitude of displacement can never be greater than the distance travelled by the body.

Consider an example:

• A boy travels 4 km towards the east and then 3 km towards the north. 
• The total distance travelled by the boy is OA AB=4 3 km =7 km, whereas the displacement of the boy in the same distance is OB= 5 km in direction 36.9° due north from east.
• Since the magnitude of displacement is the distance between the final and initial positions, It is always lesser than the magnitude of distance.
• The distance is the length of paths travelled by the body, so it is always greater than zero, i.e. it is always positive, but displacement is the shortest length of the final position from its initial position. So, it can be positive or negative depending on how the body has covered a distance.

Condition when distance equal to the displacement

However long the body is under linear motion, the displacement and the distance of the body are equivalent. Such a movement can be written as ax by c=0, that is, a straight line. 

For instance, a body tossed vertically upwards will have its magnitude of displacement and the distance the same up to its most extreme tallness. 

Remember that displacement is a vector amount, and distance is a scalar. In this way, we can’t straightforwardly say that displacement is equivalent to distance. We should compose that the magnitude of displacement is the same as the distance voyaged.

Condition when displacement can be zero even if the distance is not zero

After travelling, if a body comes back to its starting point, in this case, the distance travelled by the body will not be zero, but the displacement of the body will be zero.

Examples-(i)

• When a thing is thrown vertically upwards from a point A on the ground, after some time, it reaches back to the bottom to the same point A. 
• In this case, the displacement of the body will be zero, but the distance travelled by the body will be not zero (it is 2h if h is the maximum height attained by the body from the ground). 

Remember that displacement will not be zero when it is at its maximum height; it only becomes zero after touching back the same point on the ground from which it was thrown upwards.

Example-(ii)

•  A car moving in a circular (or closed) path reaches its original position after one round. 
• Then the displacement at the end of one round will be zero, but the distance travelled by car will be equal to the circumference of the circular path, i.e. 2πr if r is the radius of the circular path.

Conclusion

Therefore, we know from these examples and logic that whatever happens, the magnitude of displacement will always be equal to or lesser than the magnitude of distance.