Distance- Definition, Formulae, Practice problems, FAQs

Sowmya wants to get some workout in the morning, so she cycles to the playground. She starts from a point on one side of the playground, and walks till she reaches a point diametrically opposite. Let's say the time taken for this is Now she completes one full revolution around the ground by walking at the same speed. She notes down the time taken for this as and observes that is greater than Why is this so? This is because time taken She has to cover a greater distance in the second case. On the other hand, in the first case, when she covers a distance equal to the diameter of the circle, she doesn’t have to cover much distance. Now she gets on her bicycle and rides in a zigzag pattern on the ground before she reaches the other end. She notes down the time taken for this as and finds that is higher than or So the distance traveled in the third case is greater in comparison to the second or first case. Let’s see how to calculate distance and more about it!

Table of contents

Position vector
Distance and position
Displacement
Practice problems
FAQs

Position vector

A vector which is needed to define the position of an object in magnitude and direction wrt to the origin of the coordinate system is called position vector. In the following figure, the point can be written in vector form as

Distance and position

The position of a point is defined as its location in space at a certain instant of time. In the following diagram, considering as the reference point, the position of is and that of is

Distance is defined as the total path length traveled by a body in space. Consider the following example where a body starts from to and then comes back to

Then the distance traveled in this case Distance is a scalar quantity with a SI unit of meter ( The other units of distance is and Its dimensional formula is

Displacement

Displacement is defined as the change in position vector of an object. It can also be defined as the shortest distance that connects two points. Consider the following case where the position vector of a particle changes from point to point Then the displacement undergone would be

Consider a body moving in a circular path as shown in the figure. Let indicate the radius of the circle. When the body completes a full circle, to and to , then its displacement is zero and the distance is . On the other hand, when the particle moves in a half circle from to , the distance is but its displacement is

Video explanation

https://www.youtube.com/watch?v=kaLpnbmHPVY

Practice problems

Q 1.Consider a car traveling along a straight line east, then takes a U-turn and moves west. Find the distance as well as displacement of the car shown in the figure below?

(a) (b) (c) (d)

A. b

Distance is total path length, i.e

Displacement is shortest distance = , to the east.

Q 2. An athlete finishes running each round of a circular track having diameter in Calculate the distance covered and the displacement when have elapsed?

Given, diameter of the circular trackradius

Total time

Now complete rounds half a round.

For complete rounds, the displacement is zero and for half the round, the displacement

In one full revolution, the distance covered by the athlete would be equal to the circumference of the circle.i.e

In half a revolution, the distance covered would be

The distance covered after rounds would be

Q 3. A wheel having a radius of rolls forward to complete half a revolution on a horizontal ground. The magnitude of the displacement of the point of the wheel which was initially in contact with the ground would be?

(a) (b) (c) (d)

A.c

Given, radius

The horizontal distance traveled by the wheel

Now the displacement of wheel