## Distance and Displacement-

Before you go through this article, make sure that you have gone through the previous article on Distance and Displacement.

We have discussed-

• Distance is the total length of the actual path travelled by a particle during its motion.
• Displacement is a vector drawn from the initial position to the final position of the particle.

## Problem-01:

A particle covers half of the circle of radius R. Then, the displacement and distance of the particle are respectively-

1. 2πR, 0
2. 2R, πR
3. πR/2, 2R
4. πR, R

## Solution-

Consider particle moves from point A to point B covering half of the circle as shown-

### Displacement-

Magnitude of displacement of the particle

= Shortest distance between point A and point B

= Diameter of the circle

= 2R units

### Distance-

Distance covered by the particle

= Total length of the actual path

= Circumference of the circle / 2

= 2πR / 2

= πR units

Thus, Option (B) is correct.

## Problem-02:

A particle moves in a quarter circle of radius R. What are the values of distance and displacement respectively?

1. πR/2, √2R
2. πR, R
3. 2√2R, πR/2
4. πR/2, 0

## Solution-

Consider particle moves from point A to point B covering half of the circle as shown-

### Distance-

Distance covered by the particle

= Total length of the actual path

= Length of arc AB

= π/2 x R       { Using the relation θ = arc / radius }

= πR/2 units

### Displacement-

Magnitude of displacement of the particle

= Shortest distance between point A and point B

= Length of line AB

= √(OA2 + OB2)

= √(R2 + R2)

= √2R units

Thus, Option (A) is correct.

## Problem-03:

A mosquito flies from a point A(-1, 2, 3) m to a point B(2, -1, -2) m. Find the displacement of the mosquito.

## Solution-

We have-

• Initial position vector, $\overrightarrow{r1}&space;=&space;-\hat{i}&space;+&space;2\hat{j}&space;+&space;3\hat{k}$
• Final position vector, $\overrightarrow{r2}&space;=&space;2\hat{i}&space;-&space;\hat{j}&space;-&space;2\hat{k}$

Displacement vector of the mosquito $\overrightarrow{r}$ is given by-

$\overrightarrow{r}&space;=&space;\overrightarrow{r2}&space;-&space;\overrightarrow{r1}$

So, we have-

$\overrightarrow{r}&space;=&space;(2\hat{i}&space;-&space;\hat{j}&space;-&space;2\hat{k})-(-\hat{i}&space;+&space;2\hat{j}&space;+&space;3\hat{k})$

$\overrightarrow{r}&space;=&space;3\hat{i}&space;-&space;3\hat{j}&space;-&space;5\hat{k}$

Thus, displacement vector of the mosquito is-

${\color{DarkRed}\mathbf{\overrightarrow{r}&space;=&space;3\hat{i}&space;-3\hat{j}&space;-&space;5\hat{k}}}$

The magnitude of displacement is given by-

|$\overrightarrow{r}$|

= √(32 + 32 + 52)

= √(9 + 9 + 25)

= √43 m

Thus, magnitude of displacement of the mosquito = √43 m.

## Problem-04:

A mosquito flies from corner A to the corner B of a cube of side length 5 m moving along its sides as shown. Find the displacement of the mosquito.

## Solution-

Consider the origin of Cartesian coordinate system to be present at point A.

Then-

• Coordinates of point A = (0, 0, 0)
• Coordinates of point B = (5, 5, 5)

Displacement vector of the mosquito is given by-

$\overrightarrow{r}&space;=&space;(5-0)\hat{i}&space;+&space;(5-0)\hat{j}&space;+&space;(5-0)\hat{k}$

$\overrightarrow{r}&space;=&space;5\hat{i}&space;+&space;5\hat{j}&space;+&space;5\hat{k}$

Thus, displacement vector of the mosquito is-

${\color{DarkRed}&space;\mathbf{\overrightarrow{r}&space;=&space;5\hat{i}&space;+&space;5\hat{j}&space;+&space;5\hat{k}}}$

The magnitude of displacement is given by-

|$\overrightarrow{r}$|

= √(52 + 52 + 52)

= √(3 x 52)

= 5√3 m

Thus, magnitude of displacement of the mosquito = 5√3 m.

## Problem-05:

A particle is moving in a circular path of radius R. The distance and displacement of the particle when it describes an angle of 60° from its initial position respectively are-

1. πR/3, R
2. πR/6, √2R
3. πR/3, √2R
4. πR/3, √2R

## Solution-

Consider particle moves from point A to point B covering an angle of 60º from its initial position as shown-

### Distance-

Distance covered by the particle

= Total length of the actual path

= Length of arc AB

= π/3 x R       { Using the relation θ = arc / radius }

= πR/3 units

### Displacement-

In Δ AOB,

• OA = OB = R
• ∠AOB = 60°

Now,

• We know, angles opposite to equal sides are equal. So, ∠OAB = ∠OBA.
• Also, we know sum of three angles in a triangle is 180°.
• So, we conclude ∠OAB = ∠OBA = 60°.

Since all the angles in ΔAOB are 60°, so ΔAOB is an equilateral triangle.

So, all the sides length must be same.

Thus, length of line AB = R units.

Now,

Magnitude of displacement of the particle

= Shortest distance between point A and point B

= Length of line AB

= R units

Thus, Option (A) is correct.

## Problem-06:

A player completes a circular path of radius R in 40 seconds. Its distance and displacement at the end of 2 minutes 20 seconds will be-

1. 7πR, 2R
2. 2R, 2R
3. 2πR, 2R
4. 7πR, R

## Solution-

We have number of revolutions made by the player in 40 seconds = 1.

So, number of revolutions made by the player in 2 minutes 20 seconds (140 seconds)

= $\frac{1}{40}$ x 140

= 3.5 revolutions

### Distance-

Distance covered in 3.5 revolutions

= 3.5 x Distance covered in 1 revolution

= 3.5 x Circumference of the circle

= 3.5 x 2πR

= 7πR units

### Displacement-

Magnitude of displacement after 3.5 revolutions

= Diameter of the circle

= 2R units

Thus, Option (A) is correct.

## Problem-07:

Given that P is a point on a wheel rolling on horizontal road. The radius of the wheel is R. Initially, the point P is in contact with ground. The wheel rolls through half of the revolution. What is the displacement of point P?

## Solution-

Given-

• Initially, the point P is in contact with ground.
• The wheel rolls through half of the revolution.

After rolling through half revolution, point P is at the top most point of the wheel.

So, we have-

Here, d represents the displacement of point P.

Using Pythagoras theorem, we have-

d2 = (2R)2 + (πR)2

d2 = 4R2 + π2R2

d2 = (4+π2)R2

∴ d = R√(4+π2)

Thus, magnitude of displacement of point P = R√(4+π2) m.

To practice more problems on distance and displacement,

Watch this Video Lecture

Next Article- Speed and Velocity

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## Motion in One Dimension-

Before you go through this article, make sure that you have gone through the previous article on Motion in One Dimension.

We have discussed-

• In one-dimensional motion, particle moves along a straight line.
• It is also known as rectilinear motion.

Various parameters related to motion are-

## Accelerated Motion-

• The motion of a particle is said to be accelerated if its velocity changes with time.
• Example- A vehicle moving on a crowded road.
• It is a kind of non-uniform motion.

## Acceleration-

 Rate of change of velocity of a particle is called as its acceleration.

Mathematically,

### Characteristics-

The characteristics of acceleration are-

• It is a vector quantity.
• Acceleration can be positive, negative or zero.
• Acceleration is zero for a moving particle if it moves with constant velocity.
• The SI unit of acceleration is meter per second2 (m/s2).
• The dimensional formula of acceleration is [M0L1T-2].

## Types of Acceleration-

There are mainly following four types of acceleration-

1. Uniform Acceleration
2. Variable Acceleration
3. Average Acceleration
4. Instantaneous Acceleration

### 1. Uniform Acceleration-

 A particle is said to be moving with uniform acceleration if equal change in velocity takes place in equal intervals of time. OR A particle is said to be moving with uniform acceleration if its velocity changes with a uniform rate.

### 2. Variable Acceleration-

 A particle is said to be moving with variable acceleration if its velocity changes equally in unequal intervals of time or unequally in equal intervals of time.

### 3. Average Acceleration-

 The average acceleration of a particle is that constant acceleration with which the particle undergoes same change in velocity in a given time as it undergoes while moving with variable acceleration during the given time.

Mathematically,

It is the ratio of total change in velocity of the particle to the total time interval in which this change in velocity takes place.

Consider-

• At any time t1, the velocity of the particle is v1.
• At time t2, the velocity becomes v2.

Then for this interval, average acceleration is given by-

### 4. Instantaneous Acceleration-

 Instantaneous acceleration is the acceleration of particle at a particular instant of time.

Mathematically,

Instantaneous acceleration is the limiting value of average acceleration as the time interval becomes infinitesimally small.

If  Δv is the change in velocity of particle in the time interval Δt, then-

## Important Notes-

It is important to note the following points-

### Note-01:

• Acceleration is always in the direction of increasing velocity.
• Change in velocity = Final velocity – Initial velocity

### Note-02:

It is worth remembering the following relation-

### Note-03:

We know, Average Velocity

= Total displacement / Total time taken

But if the motion is uniformly accelerated, then average velocity can also be written as-

### Note-04:

• Negative acceleration is called as Retardation or Deceleration.
• It is responsible for decreasing the velocity of a particle.

To gain better understanding about Acceleration,

Watch this Video Lecture

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## Motion in One Dimension-

Before you go through this article, make sure that you have gone through the previous article on Motion in One Dimension.

We have discussed-

• In one-dimensional motion, particle moves along a straight line.
• It is also known as rectilinear motion.

Various parameters related to motion are-

## Speed-

 Rate of change of distance of a particle is called as its speed. OR Distance covered by a particle per unit time is called as its speed.

Mathematically,

### Example-

Consider-

• A particle moves in a circular path of radius 7 m.
• It starts its journey from point A and ends at the same point in time 11 seconds.

Then, Speed of the particle

= Distance covered by the particle during its motion / Time taken

= Circumference of the circle / Time taken

= (2 x 22/7 x 7) / 11

= 4 m/s

### Characteristics-

The characteristics of speed are-

• It is a scalar quantity.
• Speed is always positive. It can never be negative.
• Speed can increase or decrease with time.
• Speed is never zero for a moving particle.
• The SI unit of speed is meter per second (m/s).
• The dimensional formula of speed is [M0L1T-1].

## Types of Speeds-

There are mainly following four types of speeds-

1. Uniform Speed
2. Variable Speed
3. Average Speed
4. Instantaneous Speed

### 1. Uniform Speed-

 A particle is said to be moving with uniform speed if it covers equal distances in equal intervals of time. OR A particle is said to be moving with uniform speed if it moves continuously with constant speed.

It is worth remembering-

The direction of motion of the particle may change while moving with constant speed.

### 2. Variable Speed-

 A particle is said to be moving with variable speed if it covers equal distances in unequal intervals of time or unequal distances in equal intervals of time.

### 3. Average Speed-

 The average speed of a particle is that constant speed with which the particle covers the same distance in a given time as it does while moving with variable speed during the given time.

Mathematically,

It is the ratio of total distance travelled by the particle to the total time taken in which the distance is travelled.

If Δx is the distance travelled by particle in time Δt, then average speed is given by-

### 4. Instantaneous Speed-

 Instantaneous speed is the speed of particle at a particular instant of time.

Mathematically,

Instantaneous speed is the limiting value of average speed as the time interval becomes infinitesimally small.

If  Δx is the distance covered by a particle in the time interval Δt, then-

## Velocity-

 Rate of change of displacement of a particle is called as its velocity. OR Rate of change of position vector of a particle is called as its velocity.

Mathematically,

Its direction is same as that of displacement.

### Example-

Consider-

• A particle moves in a circular path of radius 7 m.
• It starts its journey from point A and ends at the same point in time 11 seconds.

Velocity of the particle

= Displacement of the particle / Time taken

= 0 / 11

= 0 m/s

### Characteristics-

The characteristics of velocity are-

• It is a vector quantity.
• Velocity can be positive, negative or zero.
• Velocity can increase or decrease with time.
• Velocity may be zero for a moving particle.
• The SI unit of velocity is meter per second (m/s).
• The dimensional formula of velocity is [M0L1T-1].

## Types of Velocities-

There are mainly following four types of velocities-

1. Uniform Velocity
2. Variable Velocity
3. Average Velocity
4. Instantaneous Velocity

### 1. Uniform Velocity-

 A particle is said to be moving with uniform velocity if it covers equal displacements in equal intervals of time. OR A particle is said to be moving with uniform velocity if it moves continuously in the same direction with constant speed.

It is worth remembering-

### 2. Variable Velocity-

 A particle is said to be moving with variable velocity if it covers equal displacements in unequal intervals of time or unequal displacements in equal intervals of time. OR A particle is said to be moving with variable velocity if either magnitude of velocity or direction or both changes during its motion.

### 3. Average Velocity-

 The average velocity of a particle is that constant velocity with which the particle undergoes same displacement in a given time as it undergoes while moving with variable velocity during the given time.

Mathematically,

It is the ratio of total displacement of the particle to the total time interval in which the displacement occurs.

Consider-

• At any time t1, the position vector of the particle is $\overrightarrow{r1}$.
• At time t2, the position vector is $\overrightarrow{r2}$.

Then for this interval, average velocity is given by-

### 4. Instantaneous Velocity-

 Instantaneous speed is the velocity of particle at a particular instant of time.

Mathematically,

Instantaneous velocity is the limiting value of average velocity as the time interval becomes infinitesimally small.

If  Δr is the displacement of particle in the time interval Δt, then-

## Difference Between Speed And Velocity-

Some important differences between speed and velocity are-

 Speed Velocity Rate of change of distance of a particle is called as its speed. Rate of change of displacement of a particle is called as its velocity. It is a scalar quantity. It is a vector quantity. Speed is either positive or zero. It can never be negative. Velocity can be positive, negative or zero. Speed is never zero for a moving particle. Velocity can be zero for a moving particle. Speed tells nothing about the direction of motion of the particle. Velocity tell the direction of motion of the body.

## Important Points-

It is important to note the following points-

### Point-01:

According to convention,

• If the particle moves upwards or towards right, its velocity is taken positive.
• If the particle moves downwards or towards left, its velocity is taken negative.

### Point-02:

Velocity tell the direction of motion of the particle as-

• If the velocity is positive, particle must be moving towards positive direction.
• If the velocity is negative, particle must be moving towards negative direction.

### Point-03:

• If motion takes place in the same direction, then average speed and average velocity are same.
• This is because distance and displacement are then same.

### Point-04:

• The magnitude of instantaneous velocity is always instantaneous speed.
• Instantaneous speed and instantaneous velocity differs only by direction.
• However, magnitude of average velocity is not always average speed.
• Speedometer of an automobile measures instantaneous speed of the automobile.

To gain better understanding about Speed and Velocity,

Watch this Video Lecture

Next Article- Acceleration and Retardation

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## Motion in One Dimension-

Before you go through this article, make sure that you have gone through the previous article on Motion in One Dimension.

We have discussed-

• In one-dimensional motion, particle moves along a straight line.
• It is also known as rectilinear motion.

Various parameters related to motion are-

## Distance-

 Distance is the total length of the actual path travelled by a particle during its motion.

### Example-

Consider-

• A particle moves in a circular path of radius R.
• It starts its journey from point A and ends at the same point.

Distance covered by the particle during its motion

= Circumference of the circle

= 2πR units

### Characteristics-

The characteristics of distance are-

• It depends upon the path followed by the particle.
• It is a scalar quantity.
• Distance is always positive. It can never be negative.
• Distance can not decrease with time.
• Distance is never zero for a moving particle.
• The SI unit of distance is meter (m).
• The dimensional formula of distance is [M0L1T0].

## Displacement-

 Displacement is a vector drawn from the initial position to the final position of the particle. Its magnitude is equal to the shortest distance between the initial and final position.

Mathematically, it is equal to the change in position vector of the particle.

Consider-

• A particle is initially at point A having position vector $\overrightarrow{r1}$.
• The particle moves to point B having position vector $\overrightarrow{r2}$.

### Example-

Consider-

• A particle moves along a straight line from point O to point A.
• It then travels back from point A to point B as shown-

Magnitude of displacement of the particle

= Shortest distance between its initial and final position

= 70 m – 0 m

= 70 m

Direction of displacement is towards +ve X-axis.

It is important to note that here the distance covered by the particle = 100 m + 30 m = 130 m.

### Characteristics-

The characteristics of displacement are-

• It is independent of the path followed by the particle.
• It is a vector quantity.
• Displacement can be positive, negative or zero.
• Displacement can decrease with time.
• Displacement may be zero for a moving particle.
• The SI unit of displacement is meter (m).
• The dimensional formula of displacement is [M0L1T0].

## Difference Between Distance And Displacement-

Some important differences between distance and displacement are-

 Distance Displacement It is the total length of the actual path travelled by a particle during its motion. It is the shortest distance between the initial and final positions. It depends upon the path followed by the particle. It is independent of the path followed by the particle. It is a scalar quantity. It is a vector quantity. Distance is either positive or zero. It can never be negative. Displacement can be positive, negative or zero. Distance can never decrease with time. Displacement may decrease with time. Distance between a given set of initial and final position can have infinite values. Displacement between a given set of initial and final position is unique. Distance is never zero for a moving particle. Displacement may be zero for a moving particle.

## Important Points-

It is important to note the following points-

### Point-01:

• Magnitude of displacement can never be greater than distance.
• Magnitude of displacement is equal to the distance if the particle moves along a straight line without changing the direction.

### Point-02:

When a particle returns to its initial position,

• Its displacement is always zero.
• Its distance covered is not zero.

### Point-03:

According to convention,

• If the particle moves upwards or towards right, its displacement is taken positive.
• If the particle moves downwards or towards left, its displacement is taken negative.

To gain better understanding about Distance and Displacement,

Watch this Video Lecture

Next Article- Problems On Distance & Displacement

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## Frame Of Reference-

 A frame of reference is a frame in which an observer sits and makes observations.

• A Cartesian Coordinate System is attached with the frame of reference.
• A clock is positioned in this system to measure time.
• An observer is always considered to be present at the origin O.

At any time, the position of a particle is described with the help of its three position coordinates (x, y, z).

### Position Vector-

Consider the particle is present at any point P in the space whose coordinates are (x, y, z).

The position vector of point P is a vector drawn from the origin O of the coordinate system to point P.

It is given by-

## Types of Frame of Reference-

Frame of reference can be of two types-

1. Inertial Frame of Reference
2. Non-Inertial Frame of Reference

### 1. Inertial Frame of Reference-

• A frame of reference which is either at rest or moving with constant velocity is known as inertial frame of reference.
• Newton’s laws are valid in this frame.

### 2. Non-Inertial Frame of Reference-

• A frame of reference moving with some acceleration is known as non-inertial frame of reference.
• Newton’s laws are not valid in this frame.

## Rest and Motion-

 A body is said to be at rest if it does not change its position with time. A body is said to be in motion if it continually change its position with time.

• Rest and motion are relative terms.
• When a particle moves, its position coordinates change with time.
• At one time, one or two or all three position coordinates may change.

Accordingly, we have following three types of motion-

1. Motion in one dimension
2. Motion in two dimensions
3. Motion in three dimensions

## 1. Motion in One Dimension-

 The motion of a particle is said to be in one dimension if only one out of the three coordinates specifying the position of the particle change with time.

• In such a motion, the particle moves along a straight line.
• One dimensional motion is sometimes known as rectilinear or linear motion.

### Examples-

Examples of motion in 1 dimension are-

• A car moving on a straight road
• A ball thrown vertically up
• A stone dropped into a well
• A ball dropped from a certain height above the ground

## 2. Motion in Two Dimensions-

 The motion of a particle is said to be in two dimensions if any two out of the three coordinates specifying the position of the particle change with time.

In such a motion, the particle moves in a plane.

### Examples-

Examples of motion in two dimensions are-

• Projectile motion
• Circular motion
• A carom coin rebounding smoothly from the side of the board
• Motion of a boat in a river
• Motion of an insect on a floor

## 3. Motion in Three Dimensions-

 The motion of a particle is said to be in three dimensions if all the three coordinates specifying the position of the particle change with time.

In such a motion, the particle moves in a space.

### Examples-

Examples of motion in three dimensions are-

• Random motion of a gas molecule
• A kite flying in the sky
• A bird flying in the sky

## Terms Regarding Motion-

We discuss the following parameters in detail while studying the motion of any particle-

1. Distance
2. Displacement
3. Speed
4. Velocity
5. Acceleration

## Problem-01:

“Rest and Motion are relative terms.” Justify this statement.

## Solution-

• It is rightly mentioned that the rest and motion are relative terms.
• This is because they depend on the observer’s frame of reference.
• A body at rest with respect to one observer might be in motion with respect to another observer.
• A body in motion with respect to one observer might be at rest with respect to another observer.

Consider the following examples-

### Example-01:

• A person sitting in a moving train is at rest with respect to fellow passengers.
• While person sitting in a moving train is in motion with respect to a person standing on the platform.

### Example-02:

Consider the following scenario-

• Two persons A and B are standing in a truck moving with velocity v.
• Person C is standing on the ground.

Here,

• A and B are at rest with respect to each other.
• A and B moves with velocity V in the positive X direction with respect to C.
• C moves with velocity V in the negative X direction with respect to A and B.

## Problem-02:

Is it possible to achieve state of absolute rest or absolute motion?

## Solution-

• A body at rest with respect to all other bodies is called in state of absolute rest.
• A body moving with respect to all other bodies is called in state of absolute motion.

• It is impossible to achieve the state of absolute rest.
• This is because all heavenly bodies are moving with respect to each other.

• It is impossible to achieve the state of absolute motion.
• This is because there exists no reference point which is absolutely fixed in the space.

To gain better understanding about Rest and Motion,

Watch this Video Lecture

Next Article- Distance and Displacement

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