# Motion (Kinematics) Notes for SSC CGL, CHSL & Railways 2019

# Motion

When an object changes its position with respect to its surroundings with time.

## Rest

When an object doesn’t change its position with respect to its surroundings with time.

NOTE:

- Rest and position are relative terms, i.e., an object in one situation can be at rest but in other situation same object can be in motion.
- No force is required for an object to be in uniform motion.

## TYPES OF MOTION

### 1. Rectilinear Or Translatory Motion

When a particle or a body moves along a straight path, its motion is Rectilinear or translatory motion.

### 2. Circular and Rotatory Motion

*Circular Motion*: When a body moves in a circular path.*Rotatory Motion*: When a body spins about its own axis.- Direction of motion changes at every point in Uniform Circular Motion. This direction is given by a tangent drawn at that point.

### 3. Oscillatory and Vibratory Motion

When a body moves to and from or back and forth repeatedly about a fixed point in a definite interval of time, it is said to be in oscillatory motion.

*Amplitude*: The extent to which the body moves on either side of a fixed point.*Vibratory Motion*: The oscillatory motion, where the amplitude is very small.

## One, Two and Three Dimensional Motion

### 1-D Motion (Motion in a line)

When position of an object changes only along one axis.

### 2-D Motion (Motion in a plane)

When position of an object changes along any two axis. ex: motion of a planet around the Sun.

### 3-D Motion (Motion in space)

When position of an object changes in all the 3 axes.

## Basic Terms

### Distance

Distance is the actual path traveled by a body in a given period of time.

*Odometer*– device used to measure distance traveled by the vehicle- Scalar quantity
- SI unit:
**m**

### Displacement

Displacement is the shortest distance between any two points.

- The change in the position of the object in a given period of time.
- Vector quantity
- SI Unit:
**m**

NOTE: Displacement may be positive, negative or zero whereas distance is always positive. |Velocity| ≤ Speed

### Speed

Distance traveled by a moving object in unit time interval is called speed.

speed = Distance / Time

- Scalar quantity
- SI unit:
**ms**^{-1} *Instantaneous speed*: The speed of an object at any particular instant of time. lim_{Δt→0}(Δs/Δt) = ds/dt*Uniform Speed*: If an object covers equal distance in equal intervals of time.*Non-uniform (Variable) Speed*: If an object covers unequal distance in equal intervals of time.*Average Speed*: Total Distance Covered / Total Time Taken

### Velocity (AKA Speed with direction)

The velocity of a moving object is defined as the displacement of the object in unit time interval.

velocity = displacement / time

- Vector quantity
- SI unit:
**ms**^{-1} *Uniform Velocity*: If a body covers equal displacement in equal intervals of time.*Non-uniform (Variable) Velocity*: If a body covers unequal displacement in equal intervals of time*Average Velocity*: V_{avg}= (u + v) / 2*Relative Velocity*

= V_{a}+ V_{b}(if two bodies (a and b) travel in opposite direction)

= V_{a}– V_{b}(if two bodies (a and b) travel in same direction)

### Acceleration (a)

Acceleration of an object is defined as the rate of change of velocity per unit time.

a = (v-u) / t or a = Δv/Δt

- SI Unit:
**ms**^{-2} - a is +ve, if it is in the direction of velocity and a is taken as -ve, if it is opposite to the direction of velocity.
*Uniform Acceleration*: If the object travels in a straight line & its velocity changes uniformly in equal intervals of time.*Non-uniform Acceleration*: If the object travels in a straight line & its velocity changes non-uniformly in equal intervals of time.*Average Acceleration*: Change in velocity divided by an elapsed time. a_{avg}= (V_{f}– V_{i}) / (t_{f}– t_{i})*Instantaneous Acceleration*: acceleration at any particular instant of time.

NOTE: If u > v, a is deceleration or retardation

## Equations of motion

v = u + at s = ut + 1/2 at^{2}v_{2}= u^{2}+ 2as

### Circular Motion

### Time Period

Time taken by an object to complete one revolution.

### Frequency

No. of revolutions completed by the object on its circular path in a unit time.

### Relation between Time Period and Frequency

f = 1 / T

### Angular Displacement (θ)

The angle subtended at the centre of a circle by a body moving along the circumference of the circle is called the Angular Displacement of the body.

- SI Unit:
**radian (rad)** - Angular Displacement = Arc / Radius

### Angular Velocity (ω)

Angular Velocity = Angular Displacement / time ω = θ / t

### Angular Acceleration (α)

The angular acceleration of an object moving around a circular path is defined as the time rate of change of its angular velocity.

- It occurs due to the change in direction of angular velocity. So its direction is also normal to the plane of circle in accordance with the direction change of angular velocity.
- SI Unit:
**radian / (sec)**^{2}

### Centripetal Force

When a body is in a circular motion, a force always acts on the body towards the centre of the circular path, this force is called centripetal force.

- If a body of mass m is moving on a circular path of radius r with uniform speed v, then the required centripetal force:
- F = mv
^{2}/ r

Relation between Linear and Angular Velocity

v = r x ω

Relation between Linear and Angular Acceleration

a = r x α

### Centripetal Acceleration (a_{c})

Acceleration acting on a particle undergoing a uniform circular motion towards the centre of the circle is called centripetal acceleration.

a_{c}= v^{2}/ r

### Centrifugal Force

An inertial force directed away from the axis of rotation that appears to act on all objects when viewed in a rotating frame of reference.

- Pseudo Force
- It is equal and opposite to centripetal force.

### Application of centripetal and Centrifugal forces:

- Roads are banked at turns to provide required centripetal force for taking a turn.
- The cream is separated from milk when it is rotated in a vessel about the same axis.
- The gravitational force of attraction between earth and sun acts as centripetal force.
- Orbital motion of electrons around the nucleus.
- Cyclist inclined itself from vertical to obtain required centripetal force.

### Free Falling Objects

The objects falling towards the earth under the gravitational force alone, are called free falling objects and such fall is called as free fall.

- g = 9.8 ms
^{-1} - Though the value of g is independent of free falling mass, a feather falls much slowly than a coin when released from a height. This is due to the resistance offered by air to the falling mass.
- If released at the same time in vacuum, they would reach at the earth surface at the same time.
- a +ve, if object falls vertically downward, a -ve, if object falls vertically upward
- If object is thrown vertically upward, its final velocity is zero.
- If object is thrown vertically downward, its initial velocity is zero.

### Projectile Motion

When an object is thrown obliquely near the earth’s surface, its motion on a parabolic path is known as Projectile Motion.

- The path followed by the object is called as Trajectory.

### Graphical Representation

### 1. Distance Time Graph

The change in the position of an object with time can be represented on the distance-time graph.

- The distance time graph for a body moving at uniform speed is always a straight line as distance travelled by the body is directly proportional to time.
- The distance time graph for a body moving with non-uniform speed is a curve.
- The distance time graph is parallel to time axis when the object is at rest.
- Slope of Distance Time graph gives average velocity.

### 2. Velocity Time Graph

The variation in velocity with time for an object moving in a straight line can be represented by a velocity-time graph.

- Slope of velocity time graph of moving body gives its acceleration.
- Area of v-t graph gives distance

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