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# Displacement

In physics, mechanics includes the study of the object’s motion. Kinematics is the branch of mechanics science that describes the motion of objects with the use of some terms, diagrams, graphs, numbers, and equations. These are also described by mathematical quantities which are used for determining the motion of objects. These are divided into two types that are a vector or a scalar which can be distinguished by their particular definitions which shows that Scalars quantities are easily represented by a magnitude or their numerical value while vectors quantities are described by use of both a magnitude and a direction.

Some examples of these quantities are acceleration, distance, speed, displacement, velocity etc. If we discuss distance and displacement are two quantities out of which distance is scalar quantity while displacement is vector quantity which shows the change in position of object but the distance shows the covered length by moving object. When we travel from one location to other location for reach fast on the destination, we take shortcuts making distance less so we prefer straight path over curved path. So these two quantities are related to the motion of object. Here we discuss all about the displacement.

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## Displacement Definition

Displacement is that what tells about the change in position. It is a vector and is the overall change in the position from the reference point. It is the shortest distance between the initial point and the final point. It prefers straight line path over curved path and is a vector quantity.

From these definitions, it is not yet clear what the difference between the distance and the displacement is? For understanding the difference between these quantities we must look at the following example.

Example:

A person has walked as follows (i) For the first 10 seconds, he walked to the north for 100 meters. (ii) For the next 30 seconds, he turns west and walks for 200 meters. (iii) For the next 10 seconds, he turns north and walks for 300 meters. (iv) Find the distance and displacement of the person from its original place.

Solution: As we know that the distance is total ground the person has traveled, so the distance is given by Distance = 100 + 200 + 300 = 600 meters. But the displacement is given by Displacement = 100 + 300 = 400 meters.

To understand it, we need to look at the picture below:
From the figure, if we want to find the displacement, it is the overall distance from the initial point (a) to the final point (B), which is 400 meters only, i.e. the direct distance between A and B.
From this example, we can conclude that displacement is the shortest distance between two points.

## Formula

There is no single formula for displacement, it depends on the other quantities present in the problem. Still, there are a few formulae using which we can find the displacement.
S = vt
S = ut + $\frac{1}{2}$at2
where u and v are initial and final velocity and a is acceleration.

## Distance vs Displacement

• The Distance as already discussed is the scalar quantity and it is the total path or actual path covered by the object between initial and the final point.The displacement is the shortest path between two points and it is the vector quantity.
• Both are measured in unit of lengths. Distance is generally associated with the speed of the body while displacement with the velocity of the body.
• The distance is also said to be the magnitude of displacement.

## Angular Displacement

When an object is moving in a circular or curved path, its displacement is measured as the change of angle from its initial position to the final position. This displacement is known as angular displacement. It is measured in radians.

The angular displacement is given by $\theta$ where $\theta$ is the angle between initial and final position of the object moving in Curved path.

The relation between linear displacement and angular displacement is
S = r $\theta$
where, r = radius of the curve along which object is moving

## Displacement Vector

Let us consider the motion of an object from point A to B as shown in fig.

The Actual distance covered by the object is shown by the thick line, while the light line denotes the displacement and is known as displacement vector from point A to point B.

## Resultant Displacement

The 'Resultant' term itself means the Sum or Total. So, the term Resultant displacement is the sum of all the displacements which have taken place.

## Can Displacement Be Negative?

From the above discussion we know that the displacement is the vector quantity, i.e., it has both magnitude and direction. If displacement is negative, it does not mean that the displacement is getting decreased, it means that the displacement is taking opposite direction. And hence the negative sign indicates that the object is moving in the opposite direction with respect to its initial position.

## Examples

### Solved Examples

Question 1: A car is moving with a uniform velocity of 10 m/sec. It travels to east, starting from point A, for 30 seconds, and then it turns left and moves for another 40 seconds. Find its resultant displacement?
Solution:

While traveling to the east it covered a distance of 300 meters (10 $\times$ 30 = 300) and when it turns left it is moving in north direction for another 30 seconds so the distance it covers in north direction is 200 meters (10 $\times$ 40 = 400). The figure of this motion is as shown:

We know that the displacement is the Shortest distance between two points,
Since the figure is Right angled triangle, the resultant displacement will be,
AB = $\sqrt{AO^{2} + OB^{2}}$
= $\sqrt{300^{2} + 400^{2}}$
= 500 m.

Question 2: A man is walking from his office to home. He exits the office and walks 400 m west. He then turns and walks 700 m north. Determine the magnitude and direction of his resultant displacement?
Solution:

His resultant displacement is given by :
S = $\sqrt{400^{2}+700^{2}}$
S = $\sqrt{160000 + 490000}$
S = $\sqrt{650000}$
S = 806.23 meters.

Although his total distance traveled is 1100 meters but his resultant displacement is 806.23 meters in North – West direction.

Question 3: A bus is heading city B from city A via city C. The bus started from city A and moved towards city C in south direction at the distance of 100 km. It then turns left and moves for another 200 km before reaching the city B. Find the total distance traveled by the bus and the magnitude and resultant of the displacement of the bus from the city A?
Solution:

1. Total distance traveled by the bus is D = 100 + 200 = 300 km.

2. Magnitude of the displacement is,
|S| = $\sqrt{100^{2} + 200^{2}}$,
= $\sqrt {50000}$,
|S| = 223.61 kms.

3. Direction of the resultant displacement is, the bus travels first to south and then turns left so it turns in west and hence the resultant direction from the city A is south – west.

Question 4: A boy is walking for his school and he started traveling opposite to his house, which is situated in the north direction. He traveled opposite to his house for 100 meters and then turns right and travels for 200 meters and again turns right and travels for 50 meters. Find the total distance covered by him and the magnitude and the direction of the displacement?
Solution:

1. Total distance covered by him is: 100 + 200 + 50 = 350 meters.

2. Magnitude of the displacement can be obtained by visualizing his walking,

Fig – 4
The distance from A to B is 100 meters,
from B to C is 200 meters and from C to D is 50 meters.
So, the magnitude of resultant displacement is,
|S| = $\sqrt{50^{2} + 200^{2}}$
= $\sqrt{42500}$
|S| = 206.155 m.

3. The direction of the resultant displacement is South–West.

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