Is Distance Always Positive: 9 Facts (Read This First)


9 facts related to whether distance is positive or not always are going to be discussed in this article.

We should know about distance before determining whether it is positive or not. Distance is the total path that is covered by an object. Hence distance can never be negative,the magnitude of distance can never decrease. So the answer to the question whether the distance is positive or not is yes,the distance is always positive.

Distance is a scalar quantity. Hence it has only magnitude. It does not have any particular direction. Distance can be directed in any direction. If we move to any zigzag path from one point to another then the total distance covered between the two points is known as distance.

Similarly if a person is walking around a circular park then the total distance covered by the person is the circumference of that circular park. Like displacement here the distance covered by the person is not zero because we know that distance can neither be zero nor be negative. If the radius of that circular path is R then the distance covered by that person is 2𝝅R.

Why is distance always positive?

Generally we consider the left of the origin in a coordinate axis as the negative axis and the right of the origin as the positive axis. Now whenever we are calculating the distance it means that we have to calculate the total distance between any two points.

Whether the value is in the negative axis or in the positive axis we don’t have to consider.

We only need to take the difference of the two values of lengths of these two points. In case of displacement it is mandatory to check the direction before measuring the value as it is a vector quantity. But in case of distance we can take the modulus of the negative value of the length to calculate the value of the distance as it is a scalar quantity.

Now let us take an example to clarify this concept. Say a particle is at the origin of the x coordinate axis. Now if it moves to the left upto 70 m then xi = -70 m and then if it moves 20 m to the right of the x axis,it means that xf = + 20 m. so what will be the distance covered by this particle? Answer is 50 m. why? Because here we will only consider the magnitudes not the directions i.e,positive or negative. Distance = (70 m – 20 m) = 50 m

How is distance always positive?

We all know that displacement is a vector quantity and distance is a scalar quantity. Displacement has a specific direction but distance does not have a specific direction. Basically there is a relation between distance and displacement and that relation is distance is the absolute value or magnitude of the displacement.

Now we will take a simple example to show how distance can be calculated and how distance is always positive. Say a car is moving on a horizontal coordinate axis i.e, x axis. The left side of this axis is considered as the negative axis whereas the right side of the axis is considered as the positive axis.

Now the car has moved first to the negative axis upto 100 m which means that the initial position of the car is xi = -100 m (xi < 0) and then the car has moved to the positive axis upto 50 m,hence,final position of the car is xf = +50 m (xf > 0). As here we have to calculate the value of the distance covered by that car so we will not consider the directions.

It means that we will take xi as |-100| m = 100 m and we will take xf as 50 m. hence the value of the distance will be equal to (100-50) m = 50 m. From this we can conclude that the value of the distance covered is always positive.

Can distance be negative?

In the previous section of this article we already have discussed that distance can never be negative or zero. The reason behind it is distance is basically the magnitude or the absolute value of displacement. That is why distance can never be negative, not even its value can get reduced.

But here we should know about an exceptional case. It is the measurement of distance from a mirror. So the question is why do we get a negative value of distance when we measure it from a mirror? The answer is – as we measure the distance from the pole of a mirror and in the opposite direction to the incident ray that is why we get a negative value of distance in a mirror.

Now it can be shown that distance can not be negative with the help of coordinate geometry. Say there are two points A and B on a three dimensional plane whose coordinates are A = (xA,yA,zA) and B = (xB,yB,zB). Now the distance between the two points A and B are AB = √(xB – xA)2 + (yB – yA )2 + (zB – zA)2.

Here the value of BA will also be the same as we know that distance is a scalar quantity and it has no definite direction. As we know that squaring a number can never give a negative value,hence distance can never be negative as it is the square root of sum of square terms.

Can the distance be zero?

Distance covered by a person can only be zero when the person whose distance covered is being calculated is at rest. Otherwise it is impossible that the distance covered by a moving object or a moving person is zero as distance is the total length or total path covered by them.

We can also show that the distance covered by a moving object is zero but its displacement is non zero. Suppose a car is moving around a circular track and it has ended its journey by completing one revolution around that track. In this case the initial position as well as the final position is zero,hence the displacement is zero as we know that displacement is the difference between the final position and the initial position of that car. But here the distance covered by the car is not zero as it is equal to the total path covered by the car and that is the circumference of that circular track.

In the same way we can take another example. Say a car has traveled 5 m in the east direction from a point A to another point B. After that the same car has traveled just in the opposite direction from point B to point A the same length 5 m. Can it be said that the distance traveled by this car is zero? Answer is no.

Because here direction is mandatory in case of displacement not in cse of distance. Hence in this case displacement is 5 +(-5) m = 0 but distance is 5 + 5 = 10 m.

How to measure distance?

1.Let us take an example of a hexagon first. Say it is a regular hexagon for which all the sides are equal. If we take each side of this regular hexagon is 6 cm then the total distance covered by a man is = (AB + BC + CD + DE + EF + FA) = (6 + 6 + 6 + 6 + 6 + 6) = 36 cm.

is distance always positive
Diagram for example 1

2.Let us give another example of a circular park. Say a man is running around this circular path which has a radius 15.4 m. Now what will be the total distance covered by that man? Answer is : the total distance covered by the man = circumference of that circular park =

2 x 𝝅 x R = 2 x 22/7 x 15.4 m = 96.8 m

3.Let us take another example of a moving bicycle. At first it moved 5 m,after that it moved 7 m in the same direction,then again it moved 5 m but in the opposite direction. What will be the distance covered by this bicycle?

Therefore the total distance covered by the bicycle is = ( 5 m + 7 m + 5 m) = 17 m. here the 5 m is covered by it in the opposite direction but as we know that distance is not affected by the direction of movement that is why this 5 m will be added while calculating the distance covered by the bicycle.

Diagram for the example 3

Why can the distance not be zero while the object is in motion?

For a body which is at rest the distance covered can be zero as it has not moved through any length. But in case of a body distance can never be zero as due the movement its position has changed. Though this change can be minimal, it can never be zero.

When distance can be zero?

Distance covered by a person can only be zero when the person whose distance covered is being calculated is at rest. Otherwise it is impossible that the distance covered by a moving object or a moving person is zero as distance is the total length or total path covered by them.

We can also show that the distance covered by a moving object is zero but its displacement is non zero. Suppose a car is moving around a circular track and it has ended its journey by completing one revolution around that track. In this case the initial position as well as the final position is zero,hence the displacement is zero as we know that displacement is the difference between the final position and the initial position of that car. But here the distance covered by the car is not zero as it is equal to the total path covered by the car and that is the circumference of that circular track.

In the same way we can take another example. Say a car has traveled 5 m in the east direction from a point A to another point B. After that the same car has traveled just in the opposite direction from point B to point A the same length 5 m. Can it be said that the distance traveled by this car is zero? Answer is no.

Because here direction is mandatory in case of displacement not in cse of distance. Hence in this case displacement is 5 +(-5) m = 0 but distance is 5 + 5 = 10 m.

Problem statements with solutions

  1. Mini goes to school everyday in the morning. Her school is 1.5 km away from her home. In the afternoon she comes back from school. After that she goes to a park to play with her friends which is 300 m away from her home. In the evening she comes back home. After that she goes to her tuition which is 1 km away from her house. At 10 pm she comes back home. What will be the total distance covered by her?

Answer :

HOME       ⇄   SCHOOL           (1.5 + 1.5 ) = 3 km                                                                                                                                                                                

                                            1.5 km

HOME       ⇄  PARK( 300 + 300) = 600 m = 0.6 km

                        300 m

HOME       ⇄  TUITION  ( 1 km + 1 km) = 2 km

                         1 km

Total distance covered = ( 3 + 0.6 + 2 ) = 5.6 km

  • A triangular park is there in our locality. The sides of this triangular park are 500 m ,300m and 200 m respectively. What will be the total distance covered by a child who has covered the park 3 times?

Answer :

Total distance covered by the child in 1 time = ( 500 + 300 + 200) m = 1000 m

Therefore , total distance covered by the child in 3 times = 1000 x 3 = 3000 m = 3 km

Is distance always positive
Diagram for problem statement 2

Conclusion

In this article we have discussed distance in an elaborate manner. All the questions like if distance can be negative or not, whether the distance can be zero or not, why is the distance positive always and how we can measure distance all are answered briefly. In the end there are two problems statements have given with their solutions.

Ankita Biswas

I am Ankita. I have done my B.Sc in physics honours and my M.Sc in Electronics. Currently I am working as physics teacher in a Higher Secondary School. I am very enthusiastic about high energy physics field. I love to write complicated physics concepts in understandable and simple words.

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