# Can Displacement Be Negative: 11 Facts (Read This First)

We will be discussing 11 facts related to whether displacement can be negative or not in this article.

The answer to the question can displacement be negative or not is yes. We will clarify how displacement is negative. The shortest distance between initial position and final position of a body can be defined as its displacement. Displacement is always a straight path. As displacement is a vector quantity,it has a definite direction.

This is the reason why displacement can be negative. Displacement of any moving body always depends upon the initial and final position of it not on the path that it has followed. With the help of an example we will show how displacement becomes negative. A particle is moving along the negative x axis,say it has moved up to 50 m along the negative x axis. The distance value in this case is – 50 m. Now the question arises how has it been possible?

Initially the particle was at the origin. It means that xi = 0 but finally it has moved up to 50 m in the negative x axis. It means that xf = – 50 m i.e, xf< 0. Hence according to the definition displacement = final position(xf) – initial position(xi) = xf – xi = -50 – 0 = – 50 m. In some cases displacement can be negative – this conclusion has been drawn from the example stated above.

## Why is displacement negative?

If a body has chosen to move in the negative direction i.e, it is moving either along the negative x axis, negative y axis or negative z axis in a three dimensional plane that its displacement can be said to be negative. Similarly if the initial position of a body is in a far more positive direction rather than the final position then also we can say that the displacement is negative. Now let us describe this negative displacement by a simple mathematical example.

A caterpillar is walking on a wall in the downward direction. If initially it was at 57 cm and after moving a distance up to 20 cm it stopped then what would be the value of the displacement of that caterpillar?

xi = the initial position of the caterpillar

xf = the final position of the caterpillar

Now , initially the caterpillar was at 57 cm. It means that xi = 57 cm and the caterpillar stopped at 20 cm. Hence xf = 20 cm

We know that, the displacement Δ x = xf – xi = 20 cm – 57 cm = – 37 cm

As we know that when the result is negative,it signifies a negative displacement. Hence this is the case of negative displacement.

## When displacement is negative?

There are several conditions when displacement becomes negative. First of all we need to understand one thing,that is when a body goes below the point from where it has started its journey initially then that type of displacement is referred to as negative displacement of that body.

Now one more thing we have to clarify here. When a body moves in the left side from the origin that is negative displacement and if a body falls down crossing its starting point in the downward direction then also that displacement is negative. As we know displacement is the product of velocity and time hence velocity and acceleration are the factors of displacement. Here we will talk about a few more conditions when displacement is negative.

1. When velocity = 0 and acceleration = – ve

In this case at first the body remains at rest. After that the negative direction has been chosen by it to move along. So this is a case of negative displacement.

• 2. Velocity = – ve and acceleration = +ve

Here the velocity is decreasing. So the body will move in the negative direction following negative displacement.

• 3. Velocity = – ve and acceleration = 0

In this case of negative displacement,velocity goes on decreasing and the body moves in the negative direction.

• 4.Velocity = – ve and acceleration = – ve

In this case also the body moves in the negative direction following negative displacement.

## How is displacement negative?

Let us take a simple example to understand how displacement is negative. Say a car is at rest at a point P. now it starts to move in the right side up to point Q which is at 10 m right from the point P,after that it again starts moving in the opposite direction that is towards point P and reaches point P. The car has a zero displacement in this case. Because at first it moved 10 m on the right side and then again 10 m on the left side. Hence  displacement of the car = PQ = 10 – 10 = 0 m.

Now if the car starts moving in the left direction towards R then the displacement will be negative. Certainly the question arises that why? Because we have taken the displacement in the right side of point P as the positive displacement,hence the displacement along the left side of point P should be negative. Say R is 5 m away from P on the left side. Hence PR = -5 – 0 = – 5 m

## When displacement is not negative?

There may be two cases when displacement is not negative. The first case is of zero displacement and the second case is of positive displacement. Now let us discuss these two cases:

1. Zero displacement
2. Positive displacement

#### 1. Zero displacement

This is the case when the initial position and the final position of the body is superposed with each other. If xi denotes the initial position of the body and xf denotes the final position of the body then the displacement will be = Δ x = xf – xi

In this case xi = xf hence Δ x = xf – xi = 0

Example of zero displacement is if a person starts moving from a position on a circular park and after some time gets back to the same position,then her initial as well as the final position are the same. That is why her displacement is zero.

#### 2. Positive displacement

In the positive direction if the final position of a body is far away from the initial position that case would be considered as the case of positive displacement. In this case xf> 0 and xi = 0 or xi < 0 or xi >0 ( but must have a smaller positive value than xf). A body that is continuing its motion along the positive x axis can be considered as an example of positive displacement.

## How can displacement be positive or negative?

1. Positive displacement

In the positive direction if the final position of a body is far away from the initial position that case would be considered as the case of positive displacement In this case xf> 0 and xi = 0 or xi < 0 or xi >0 ( but must have a smaller positive value than xf). A body that is continuing its motion along the positive x axis can be considered as an example of positive displacement.

• Negative displacement

First case

This is the case when the initial position of the body is far away from the final position in the positive direction. xi>0 and xf> 0

Δ x = xf – xi < 0

Second case

There can be another case of negative displacement when the initial position of a body is in the positive direction ( positive x axis) and the final position is at the starting point ( origin). xi> 0,xf = 0, Δ x = xf – xi < 0

Third case

When the initial position of the body is at the origin and the final position is along the negative x axis of a coordinate system. xi = 0,xf < 0, Δ x = xf – xi < 0

Fourth case

When the initial position of the body is along the positive x axis and the final position is along the negative x axis. xi>0,xf<0,

Δ x = xf – xi < 0

## Can displacement be positive?

This is the case when the final position of the body is far away from the initial position in the positive direction. There must be another case.  if a body falls down but is not able to cross its starting point in the downward direction then also that displacement is positive. In this case xf> 0 and xi = 0 or xi < 0 or xi >0 ( but must have a smaller positive value than xf). Example is a particle is moving along the positive x axis.

## Why is displacement positive?

There are several conditions when displacement can be positive.

1. When the final position of the body is far away from the initial position of the body in the positive direction,then that displacement is positive displacement.

xi> 0, xf> 0 Δ x = xf – xi > 0

• When the final position of the body is far away from the initial position in the positive direction and the initial position of the body is at the starting point,then this is a case of positive displacement. xi = 0, xf > 0, Δ x = xf – xi > 0
• When the final position of the body is in the positive direction whereas the initial position of the body is in the negative direction. xi< 0, xf > 0

Δ x = xf – xi > 0

• When the final position of the body is at the starting point and the initial position of the body is in the negative direction. xi < 0, xf = 0

Δ x = xf – xi > 0

## Examples of negative displacement

There are several examples of negative displacement.

1. A particle is moving along the negative x axis or negative y axis or negative z axis. The displacement of this particle is an example of negative displacement.
2. Nili has thrown a stone in the upward direction. The stone has reached 20 m in the upward direction and then starts falling to the ground. If it has reached the ground which is at 30 m from the point of throwing in the downward direction then what will be its displacement? Here xi = 20 m > 0  and xf = -30 m < 0

Δ x = xf – xi = – 30 – 20 = -50 m < 0. Therefore, the displacement of the stone can be referred to as a negative displacement.

## Examples of positive displacement

1. A particle is travelling along any of the orthogonal axes in the positive direction i.e, positive x axis or positive y axis or positive z axis. The displacement of this particle is an example of positive displacement.
2. Let us take a car that has chosen to move along the negative x axis. At first it has moved up to 10 m then again it has started moving towards its right and reached to 30 m in the positive x axis. What will be its displacement?

The initial position of the car is, xi = – 10 m < 0

The final position of the car is, xf = 30 m > 0

Therefore, displacement of the car is, Δ x = xf – xi

= 30 – (-10) m

= (30 + 10) m

= 40 m

## Problem statement with solution

1. 1.      A particle has chosen to move along a vertical axis which is the y axis of a coordinate system. It has started to move from the point A along the negative y axis. Point A is at 5 m in the negative y axis. After that this particle has reached point which is along the positive y axis at a distance of 5 m. then again the particle has moved to point C which is along the negative y axis at a distance of 5 m and from there the particle has moved to point D which is along the positive y axis at a distance of 5 m. then again this particle has moved to the point E along the negative y axis at a distance of 5 m and at last it has reached point F. what will its total displacement?