**Introduction:**

A frame of reference is applied in Physics mainly to understand the motion of the objects in that particular frame and related to another frame in the reference. If we consider a reference frame to understand the motion, it goes like this.

For example, a car is moving at the speed of 35 m/s, and an observer B is sitting on the bench in the park. For the observer, the car is moving at the speed of 35 m/s, but for the people in the car, it seems like they are at rest and all the things outside are moving backwards.

Therefore, to avoid such confusion, we use a frame of references. Here, we have two frames of reference the person in the car and the person on the bench.

**Explanation:**

**Let us Consider the Following Figure.**

If we ask A the velocity of B, he will say it is not moving or is at rest. But if we ask the question to C, he will say that B has a velocity V in the positive side X direction. So, before specifying the velocity, we have to determine which frame we are in. We need to find a frame of reference.

Types of Frames of References

- Inertial Frame of Reference

- Non-Inertial Frame of Reference

**Inertial Frame of references**

An inertial frame of reference is when force is not acting, and the body is in constant motion. Imagine a car which moving at a constant speed, and no external force is acting upon it. We see that car is moving at a constant speed. But we ignore that the car is experiencing some force as the Earth is constantly moving and the car is moving on the Earth surface. We can say that the force of the Earth rotation is not applied to the car, and the car is moving with a constant speed.

**Note:** In the inertial frame of reference, Newton’s laws hold perfectly. Means Newton’s can be applied to know more about the motion.

**Non-Inertial Frame of references**

- We can define a non-inertial frame as an accelerated frame for an inertial frame of reference. Newton’s law will not hold in these frames. If we consider the inertial frame of reference as car and the person in the car becomes the non-inertial frame of reference as the frame of reference moves with him faster than the inertial frame of reference. Important to understand that acceleration is the crucial component for the non-inertial frame of reference.
- If a frame of reference moves faster than the inertial frame of reference, it is said to be a non-inertial frame of reference.
- Note: In a non-inertial frame of reference, cannot apply Newton’s laws.

**Illustrating**** Motion:**

Because you need to establish a frame of reference to study motion, diagrams and sketches are critical tools. Charts show how the object’s position changes about a stationary frame during a particular or several time intervals. When comparing the object’s position in each series of pictures, you can determine whether the object is at rest, speeding up, slowing down, or traveling at a constant speed.

**Relative motion:**

Let us consider two persons are traveling in the car. Person A observes that person B is at rest and not moving but traveling in the car.

Let us consider another person, C as the observer and observes the car moving from his frame of reference and the person can see that person A and B are moving.

Person B is at rest as person A, but as per person C, person B is moving. We can say that persons A, B and C are in relative motion.

There are so many examples in our day-to-day life. There are so many instances where we can find relative motion.

Definition of Relative motion: The motion of a moving object with the object which is not moving when observed from the frame of reference of the body that is moving is called a relative motion.

What if we have so many objects moving in one reference frame? What can be done with those types of situations?

**Relative Velocity:**

Relative velocity is a phenomenon when there are a more significant number of objects moving. We must understand that the objects may be moving in the same or the opposite direction, but consider that the bodies are moving and moving with velocity.

Let’s take an example and understand the relative velocity, consider a boat is moving in a river.

- The boat is moving.
- The river is also moving. We can say that the river has a flow.

Consider that the velocity of the boat is _{VA}_{ }and the velocity of the river is v_{b.} Therefore,

The velocity of the boat concerning the river is

_{Van = VA }– v_{b}

The velocity of the river concerning the boat is

V_{BA} = v_{B}_{ }– _{VA}

On comparing both the equations, we find that the values are the same.

Mathematically we can write

Iv_{ba}I = Iv_{ab}I

**Relative Acceleration:**

If the bodies are moving with the same or constant acceleration, then they said to be in relative acceleration

Therefore, relative acceleration is given by

a_{AB} = a_{A}– a_{B}

There are four types of problems that usually appear in a relative motion.

- Minimum distance or collusion

- River and boat problems

- Aircraft and wind problems

- Rain and raindrop problems

**Minimum Distance or Collusion Problems:**

Let us consider two cars, A and B, moving in online with constant acceleration and velocity. We can find the minimum distance and collusion of the cars using the principle of relative motion.

**Solved Examples:**

Let us consider two cars, A and B, are moving with a constant acceleration of 4 m/s^{2} and a constant velocity 6 m/s with t = 0 s. Both cars are 10 m apart. car A is behind car B. Find the time when car A overtakes car B.

Given u_{A}_{ }= 0, u_{B} = 6m/s, a_{6} = 4 m/s^{2}

u_{AB} = u_{A} – u_{B} = 0-6 = ^{–}6 m/s

a_{AB} = a_{A} – a_{B} = 4-0 = 4 m/s^{2}

We know S = ut + ½at^{2}

10 = – t + ½ (4)(t^{2})

10 = -t + ½ (4)(t^{2})

2t^{2} – t – 10 = 0

t = t ± √1+80/4

t = 1 ± √81/4

t = 2.53 s

**River and Boat Problems:**

Remember three things while solving river and boat problems

- Absolute Velocity of river = V
_{r}

- The velocity of the person on the boat concerning the flow of the river or still water = V
_{br}

- Absolute velocity of person on boat = V
_{b}

V_{br} = V_{b}– V_{r}

If we derive the components of the velocity then,

V_{bx} = V_{r} – V_{br}_{ }Sinϴ

V_{by} = V_{br}Cosϴ

The time taken by the person in the boat to cross the river is given by

T = W/ V_{br}_{ }Cosϴ (w=width of the river)

The displacement (x) along the x-axis is given by

X = (v_{r}– v_{br}Sinϴ) W/ V_{br}Cosϴ

Condition for the person on the boat to cross the river in the shortest amount of time is given by

t_{min} = W/V_{br}_{ }(cosϴ = 1)

Condition for the person on the boat to reach the point exactly opposite to his initial point is given by

Here x = 0

Therefore,

V_{r} = V_{br}Sinϴ

**Example:**

The river of width 30 m has a velocity of 2 m/s. A person rowing the boat with the velocity of 5 m/s at an angle of 30 degrees.

Find the time it has taken to cross the river.

Drift(x) of the person on the boat while crossing the river

a) Time taken to cross the river t = w/vbrsin∅ = 30/2.5 = 12 s

b) Drift along the river ∂ = (5.5)(t) = 5.5(12) = 66 cm

**Air and Wind Problems:**

It is very similar to the boat and river problems. The only difference we have is steering velocity and the velocity of the aeroplane with wind and still air, which means V_{br} becomes V_{aw}.

Then,

V_{aw} = V_{a}– V_{w}

The time taken by the aeroplane is

T= AB/V_{a} where, AB is the initial point to the final point.

**Example:**

An aeroplane is flying with a speed of 400 m/s in the still air. Consider the direction of the wind from the north to south with the speed of 280 m/s, and the distance between the initial and the final point of the aeroplane travelled is 1000 km (1000000 m)

Find the time taken by the flight to fly from the initial from the final point.

According to sin a law of triangle

CB/sin45o = AC/sinα

Sinα= ( [Equation]Sin 45^{o})

(280/400) 1/√2 = 1/2

Sinα= ½ =α = 30^{o}

The steering direction of pilot (45^{o} + α) = 75^{o}

V_{a} = 400/Sin45o

V_{a}= 400 x √2 = 560 m/s

∴ Time taken T = AB/Va = 1000000/560 = 17855

17855 = 30 minutes

**Rain and raindrop problems:**

Here, we have

V_{r}= Velocity of rain

V_{p}_{ }= Velocity of person

V_{ip} = Velocity of rain with the person

Then,

V_{rp}= V_{r}– V_{p}

**Example: **

A man is walking in the rain with the speed of 3 m/s in the west direction, and it is raining vertically downwards with the speed of 4 m/s. Find the direction of the person’s umbrella so that the rain doesn’t wet him.

V_{mr}_{ }= V_{r}_{ }– V_{m}

V_{rm} = 4 – 3 = 1 m/s

tanϴ = ¾

ϴ = tan^{-1} (¾) = 37^{0}

The direction of umbrella should be 37^{0 }east

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