# Motion

## Motion :-

Motion is when an object changes its position with respect to time. For example, if a car is moving, then it changes its position with respect to time. This is called motion.

## Quantity :-

A quantity is used to quantify any element based on factors like magnitude, direction, etc. It is of two types :-

1) Scalar quantity

2) Vector quantity

Scalar quantity :- it is a quantity which is only dependent on magnitude. Magnitude is a word used for the amount. Also, these quantities are dependent on scalar laws of addition. It can never be negative or 0. For example, distance, speed, work, energy.

Vector quantity :- it is a quantity which is dependent on magnitude and direction. It is also dependent on vector laws of addition and vector algebra. It can be positive, negative or 0. For example, displacement, velocity, force, weight.

## DISTANCE AND DISPLACEMENT :-

**Distance-** it is the quantity which is used to quantify the amount of ground the object has covered. So, this can never be negative or 0 as it keeps on increasing. For example, if a bike is covering 5km and stops for a while and again continues to cover another 8km, then you can see that the total distance is 13km and it did not become negative or 0. Hence, it always increases. This is a scalar quantity. Its SI unit is meters(m).

**Displacement-** It is the quantity to capacify the shortest distance between the initial position and the final position of the object. It is a vector quantity as displacement is always said with direction. For example, 5km due north. It can be 0 when the initial position and the final position of the object is the same. It is negative when it moves downwards or leftwards. Its SI unit is meters(m).

* NOTE-* when an object is travelling continuously in a straight line without changing its direction, only in this case the object will have equal distance and displacement. Also, magnitude of displacement is always lesser or equal to the distance magnitude. You can view the below pictures for a better understanding.

## Speed and Velocity :-

**Speed**- it is a quantity which is used to measure the distance covered per unit time.

**Mathematically, it is given by- **speed = distance/time( only can be used when motion is uniform)

It is a scalar quantity because distance is a scalar quantity. Its SI unit is meters/second(m/s) because SI unit of distance and time is meters(m) and seconds(s), respectively.

### Types of speed :-

1.- **Based on uniformity**- Uniform and Non - Uniform speed

**Uniform**- it is when an object covers equal distances in equal intervals of time. For example, when a car covers a distance of 10m is 2s and 20m in 4s, then the car is said to have uniform speed.

**Non - uniform-** it is when an object covers unequal distances in equal intervals of time or equal distances in unequal intervals of time, then the body is said to have non-uniform speed.

2.-** Based on calculating the average or speed at any instant :-**

**Average speed**- it is the total distance covered by the object upon the total time taken to complete the journey. This speed gives us a rough idea on the overall speed which it was travelling with during the whole journey.

**Mathematically, it is given by- **Average speed = Total distance/Total Time

**Instantaneous speed- **it is the speed measured at any particular instant of the whole journey. For example, when an object is supposed to travel under 60kmph according to the speed limit and it violates the rules, then the police can file a case by referring to the speed at that instant. This is called instantaneous speed.

## Velocity :-

It is the displacement covered per unit time or it is the distance covered per unit time with direction. Its SI unit is m/s. It is a vector quantity because displacement is a vector quantity. Also, velocity can be 0 when displacement is 0 and can be negative when the object is slowing down.

**Mathematically, it is given by-** Velocity = Displacement/Time( only can be used when motion is uniform)

### Types of velocity :-

1.- **Based on uniformity- ** Uniform and Non-Uniform Velocity

**Uniform-** it is when an object covers equal displacements or distances with direction in equal intervals of time.

**Non-uniform**- it is when an object covers unequal displacements in equal intervals of time or equal displacements in unequal intervals of time.

2.- **Based on the average and instant :-**

**Average velocity**- it is the total displacement upon the total time of the journey. This is used to get a rough idea on how fast the object is moving during the whole journey.

**Mathematically, it is given by**- Average velocity = Total Displacement/ Total Time

**Instantaneous velocity**- It is the velocity at any instant of the whole journey.

## Acceleration :-

It is a phenomenon in which an object changes its velocity per unit time to increase its speed or decrease its speed. For example, if an object moves initially with a velocity of 5m/s and suddenly changes to 24m/s in just 2 s because the driver is in a hurry, then the body is said to be accelerating. Its **SI unit is m/s²**.

**Mathematically, it is given by**- A = V-U/ T (a - acceleration, v - final velocity, u - initial velocity )

### Types of acceleration :-

1.- **Uniform acceleration**- When an object changes its velocity equally in equal amounts of time, then it is said to have uniform acceleration.

2.- **Non uniform acceleration**- When an object changes its velocity unequally in equal amounts of time, then it is said to have non uniform acceleration.

**NOTE-** When the velocity of the body is uniform or constant then the acceleration of the body is said to be 0.

## Equations of Motion :-

**1.-** These are some equations which are very important and are used when some of the data in the question is not given and you have to solve for one of the quantity.

**2.-** These can be used only when the body is having uniform or constant acceleration.

There are 3 equations of motion along with their derivation. They are:-

**1.-** v = u + at

**2.-** s = ut + 1/2a(t)²

**3.- **v² - u² = 2as

(v is final velocity, u is initial velocity, t is time, a is acceleration, and s is displacement)

We can derive the following equations using 2 methods. They are :-

1.- mathematically 2.- graphically

### 1.- Mathematically

we know that acceleration is v-u/t, so by rearranging, we get

a=v-u/t

v = u + at

Now, we know that the second equation of motion is a position time equation. This means that the final velocity term will not be there. We also know that when velocity is uniform, we can use distance is the product of average velocity and time.

**NOTE-** When the velocity is uniform in the whole journey, to find out average speed, we can use mathematical average.

s = u + v/2 * t

Now, from the first equation of motion, we know that v = u + at

So, we substitute the value of v

s = u + u + at /2 * t = 2u + at/2 * t

This implies that

s = (u + at/2) *t = ut + 1/2 at^2

Now, we know that the third equation of motion is a position velocity equation. So, the 't' term will be omitted.

v = u + at

v^2 = u^2 + 2uat + a^2t^2 ( squaring both sides)

v^2 - u^2 = 2uat + a^2t^2

(v^2 - u^2)/2 = (2uat + a^2t^2)/2 ( dividing by 2 both sides)

(v^2 - u^2)/2 = {a(2ut + at^2)}/2

(v^2 - u^2)/2 = {a ( ut + 1/2 at^2)}

(v^2 - u^2)/2 = ( as)

v^2 - u^2 = 2as

## SLOPE CONCEPT :-

IN ANY GRAPH, WE CAN SEE A LINE PASSING THROUGH THE ORIGIN. THAT IS CALLED THE SLOPE. AND IT IS ALWAYS THE CHANGE IN Y AXIS DIVIDED BY THE CHANGE IN X AXIS.

## Derivation of Equations of Motion by Graphical Method :-

Now, we can see a velocity time graph. And we know that the slope of a velocity time graph is acceleration because change in velocity upon change in time is acceleration according to slope concept. Now, let us say that the object moves from point A with an initial velocity of u and reaches a point B with a final velocity v. Now, slope AB= B-D/ C -O

a = v -u/t

From rearranging, we get** v - u = at, v = u + at.**

Now, the second equation of motion is position time equation. And we know that distance is velocity times time. Hence, it is area under the curve.

So, we can get the value by getting the areas of triangle ABD and rectangle ADOC. So, area of triangle is 1/2 b * h

= 1/2 * AD * BD = 1/2 * v- u * t

Now, we know from the first equation of motion that v-u = at

So, 1/2 * at * t = 1/2 at²

Now, the area of a rectangle = l * b = t * u = ut

So, total area is ut + 1/2 at²

s= ut + 1/2 at²

Now, the third equation of motion is a velocity displacement equation. So, area of trapezium AOBC is the displacement

So, area of trapezium = 1/2 * height * ( sum of parallel sides)

s = 1/2 * OC * ( BC + OA )

s = 1/2 * t * ( v + u )

Now, from the first equation of motion, we know that

v= u + at

t = v-u/a( we are doing this to get rid of the variable called time)

By substitution, we get

s = 1/2 * (v-u/a )(v + u)

s = (v - u )(v + u)/2a

s = (v² - u²)/2a (a - b)(a + b) = a² - b²

v² - u² = 2as

**NOTE-** IF YOU FIND THE AREA UNDER THE CURVE BY ADDING THE AREAS OF THE TRIANGLE AND RECTANGLE, WE GET THE SECOND EQUATION OF MOTION AND IF WE FIND THE AREA OF THE WHOLE TRAPEZIUM, WE GET THE THIRD EQUATION OF MOTION

## CIRCULAR MOTION :-

When a body travels in a circular path and changes its position with respect to time, it is called circular motion. Also, the velocity is never constant because the direction always changes.

Types of circular motion

**Uniform-** When the speed is constant and the velocity is variable.

**Non - uniform-** When the speed and the velocity is variable.

The formula to calculate the speed of a circular path is given by :-

Distance/ time = 2πr/t