## Introduction to Mechanics :

- Mechanics is a branch of physics that deals with ordered(A sequence or arrangement of) motion of particle or a system of particles. Here after, ordered motion is simply referred as motion.
- Now we are going to learn Newtonian mechanics or Classical mechanics which is based on Newton's laws of motion.
- Newtonian mechanics gives correct results for macro particles(whose speed is far less than the speed of light in vacuum and mass is much greater then the mass of an atom.
- In Newtonian mechanics, mass and time are absolute ( same for any observer ).
- If the speed are comparable with speed of light in vacuum and masses are much greater than the mass of an atom, then the relativistic mechanics provides correct results. It was developed by the great scientist "Einstein".
- In relativistic mechanics, it is concluded that mass and time are not absolute. They are relative.
- If the masses involve are comparable with the mass of atom, then the quantum mechanics provides correct results.

## Reference frame

In physics, all observations are to be made from a particular reference frame. For the time being, just assume that reference frame is the background from which observations are made.

Example : A black board appears to be at rest to boy sitting in class room. If the observes the same black board from the moon, it appears to be in motion since the earth is in motion with respect to the moon.

Example : A black board appears to be at rest to boy sitting in class room. If the observes the same black board from the moon, it appears to be in motion since the earth is in motion with respect to the moon.

Rest and Motion :

If in a reference frame, the position of a particle doesn't change with time, then the particle is said to be at rest with respect to that reference frame. If the position of particle changes continuously then the particle is said to be in motion with respect to that reference frame.

Rest and Motion are relative

Example : Suppose A and B are two persons sitting in a moving train. If you ask person A, whether B is moving or not, he says "no". However a person C who is standing outside the train answers the same question saying that V is moving along with the train.

## Translatory motion

Motion of an object from one position to another position is called translatory motion.

(simply motion of particle is a translatory motion).

(simply motion of particle is a translatory motion).

Kinematics of trajectory motion :

In this branch of physics, motion is described without any reference to its cause. To describe a motion we state a mathematical quation which gives how the position of a particle is changing with time.

Path or Trajectory :
It is the line joining the successive positions of a particle in motion.
motion of straight path is called rectilinear motion & motion of curved path is called curvilinear motion. Example : If a particle is moving along a circular path then its motion is curvilinear and it is called circular motion. |

- Dimensions of motion is based on the number of coordinates used to specify the position of a particle completely
- One dimensional motion require just one coordinate to specify position of particle completely
- Two dimensional motion require two coordinates to specify position of particle completely
- Three dimensional motion require three coordinates to specify position of particle completely

Path is relative

Distance : Length of the path is called distance.

- SI unit : meter (m).
- Dimensional formula : [L]
- It is scalar

Shape and length of a path are relative. Hence distance is relative.

## Displacement

Magnitude : Irrespective to the shape of the path, length of the line segment joining the initial and the final positions of the particle gives the magnitude of the displacement.
Direction : Irrespective of the shape of the path, direction of the ray from the initial position to the final position gives the direction of the displacement.
- SI unit : meter (m).
- Dimensional formula : [L].
- It is vector.
Displacement is relative |

## Speed

In kinematics, the speed of an object is the magnitude of its velocity (the rate of change of its position)

Speed is relative

Average speed :

Distance covered per unit time is called average speed. If a particle covers a distance "s" in a time interval "t", then the average speed of particels is

Instantaneous speed :

Speed of particle at an instant is called instantaneous speed.

Uniform Speed :

Is the speed of a particle is same at every instant in an interval of time, then the speed of the particle is said to be uniform in that interval.

- Here the particles cover equal distance in equal intervals of time, however small they may be.
- when the speed of a particle is uniform and is equal to "v" then its average speed over any interval of time is equal to "v" and the instantaneous speed at any is also equal to "v"

## Velocity

In kinematics, velocity is the rate of change of the position of an object, equivalent to a specification of its speed and direction of motion.

Average Velocity :

Displacement per unit time is called average velocity.

Its direction is equal to the direction of the resltant displacement

Instantaneous velocity :

Note : Direction of instantaneous velocity is called the direction of motion at that instant.

Uniform velocity :

If the velocity of a particle is same at every instant in an interval if time then it is called uniform velocity.

- In this case, the particle covers equal distance in a particular direction in equal intervals of time however small they may be
- In this case the path must be a straight line.
- It is a vector. Its direction is along the straight line in which it is in motion.
- only in this case, the particle is said to be in uniform motion.

Change of velocity :

Velocity of a particle changes when

- Its magnitude is changed (or)
- Its direction is changed (or)
- Both magnitude and direction are changed.

Change in velocity is absolute in an inertial frame

Steps to follow in sign convention :

- Assume a direction as positive direction.
- Write all the known qualities with proper signs.
- Write the unknown quantity with positive sign.
- Substitute the above value in the equation.

## Acceleration

In physics, acceleration is the rate at which the velocity of a body changes with time.

Average Acceleration :

Change in velocity per unit time is called average acceleration.

- If the velocity of a particle changes from u ⃗ to v ⃗ in an interval of time "t", then its average acceleration.

- Its direction is equal to direction of ∆v ⃗ , that is direction of change in velocity.

Instantaneous acceleration :

Acceleration at an instant is called instantaneous acceleration.

- Its direction depends on the given situation.

Uniform acceleration :

If the acceleration of a particle is the same at every instant in a given time interval, then its acceleration is called uniform acceleration in the that interval of time.

- Its direction is equal to the the direction of ∆v ⃗

Acceleration is absolute in an inertial frame.

Path of particle moving with uniform acceleration :

__Path is a straight line__when the directions of the initial velocity and acceleration are parallel to each other.

- If the directions of the initial velocity and acceleration are same, the magnitude of the velocity of the particles increases continuously with time.
- If the direction of the initial velocity and the acceleration are mutually opposite the particles is said to be moving with retardation or deceleration. In this case the magnitude of the velocity of the particle decreases continuously until it becomes zero. After that it reserves its direction of motion and then onwards the magnitude of velocity increases continuously.
__Path is a parabola__when the initial velocity is not para;;e; to the acceleration.

## Equations of motion :

Consider a particle moving along straight line with a uniform acceleration a ⃗. If its velocity changes from u ⃗ to v ⃗ in an time "t", then its acceleration is given by

Important : Sign convention must be used while using the scalar formula

Derivation :

From the definition of acceleration we know that " v = u+at ". It is a linear equation. Hence its average velocity over a time interval "t" is given by

From the definition of average velocity, we know that displacement is given by

from equations (1) & (2)

If we substitute " v = u+at " in the above equation then we get

similarly if we substitute u = v-at, we get

In this manner if we substitute t = (v-u)/a, we get

Important : sign convention is must while using all of these equations

Note : In all of the above formula "s" represents total displacement in the time interval "t" from the initial instant

Note : In all of the above formula "s" represents total displacement in the time interval "t" from the initial instant

Displacement in "n"th second :

Consider a particle moving along a straight line with a uniform acceleration a ⃗. Let its total time of journey be "n" second and its initial velocity be u ⃗.

finally we get

because v = u+an ; again we get

## Acceleration due to gravity (g)

Any object on the earth is pulled towards its center by the earth. If no other force acts on the object the object moves with an acceleration called acceleration due to gravity and is represented by "g".

Magnitude :

Note : The value of "g" decreases with increase of height, as well as depth

Direction :

At a point on the earth we assume that ground is flat and we say that it is the vertically downward direction.

Note : the direction of "g" is always towards the center of the earth, and we say that it is in the radially inward direction.

Note : the direction of "g" is always towards the center of the earth, and we say that it is in the radially inward direction.

Acceleration due to gravity is independent of

- Direction of motion and velocity of the object.
- Mass of the object.
- Shape & size of the object.

Dropping of an object :

- If a person at rest drops an object, its initial velocity will be zero and it starts acceleration downwards with an acceleration a = g downwards.
- If a person, moving with velocity "v" in the vertically upward direction, drops an object then its initial velocity is u = v and acceleration is a = -g downwards.
- If a person, moving with a velocity "v" in downward direction, drops an object then its initial velocity, u = +v and it start acceleration downwards with an acceleration a = +g download.

Projection : ( Projection (or) throwing an object )

when a person throws an object, its initial velocity depends in the velocity of hand at the moment he releases the object.

Consider an object projected vertically upwards with an initial velocity "u" from the ground.

- Its velocity decreases continuously while it is moving vertically upwards and finally it will become zero.
- Its velocity at the maximum height is zero.
- After reaching maximum height, it starts falling. its velocity increases in the downward direction and finally it reaches the ground with the velocity "u" which is magnitude of initial velocity of projection.
- Throughout the upward motion, at the maximum height and downward motion its acceleration is a(=g) downwards.

Time of ascent :

The time taken by the object to reach the maximum height which is called time of ascent and is given by v = u+at.

If we assume vertically upward direction as positive direction,

- Initial velocity u = +u
- Final velocity during upward motion, v = 0, acceleration a = -g

Maximum height :

Time of descent :

The time taken by the object to reach the ground from the maximum height is called time of descent and is given by v = u+at.

During the downward journey.

- Initial velocity u = 0
- Acceleration a = -g
- Final velocity v = -u

Time of ascent is equal to time of descent

Time of flight :

The total time taken by the vertically projected object to reach the maximum height and then falling freely to touch the initial position is called time of flight.

Time of flight is the sum of time of ascent and time descent

Time of flight is the sum of time of ascent and time descent

by adding the values of time of ascent & time of descent we get