__REST, MOTION AND REFERENCE FRAME__

__REST, MOTION AND REFERENCE FRAME__

**Kinematics **is the branch of physics that deals with motion of bodies without inquiring its cause (the concept of forces).

** A Particle **is defined as matter of infinitesimally small size. Thus a particle has only a definite position but no dimension.

A certain amount of matter limited in all directions with finite size, shape occupying some definite space, is called a **Body**.

**Rest**

*A body is said to be at rest when it doesn’t changes its position with time. Example- A blackboard sticking to a wall for some time is said to be at rest when it doesn’t move from its position.*

**Motion**

*A body is said to be in motion if the position of the body continuously changes with time. Example- Movement of a car w.r.t. an observer at rest.*

**Frame of Reference :**

*To locate the position of a body relative to the reference body a system of coordinates fixed on the reference body is constructed. This is known as reference frame.*

If two cars A and B move side by side in same direction with same speed, it would appear to the passengers of the cars that they are mutually at rest. Obviously, B is at rest relative to A. The reverse is also true.

**Absolute rest or absolute motion is undefined. Motions are relative.**

** NOTE: Description of state of motion of a particle requires a set of axis, w.r.t to which the state must be specified, otherwise it won’t make any sense. The inherent meaning of this statement is that we need a reference frame to determine whether a body is at rest or in motion.*

__IMPORTANT TERMS AND DEFINITIONS__

__IMPORTANT TERMS AND DEFINITIONS__

Rectilinear Motion | A body is said to have a rectilinear motion when any two particles of a body travel the same distance along two straight parallel lines. |

Position vector | It describes the instantaneous position of a particle with respect to chosen frame of reference. It is a vector joining the origin of a particle. For one dimensional it’s defined as : r = xî For two dimensional : r = x î + y ĵFor three dimensional : r = x î + y ĵ + z ƙ |

Distance | The total path length covered by the particle in a given time interval is known as distance. |

Displacement | The vector joining the initial and final positions of the particle during a given time interval is called displacement. It is the change in position of a moving object. |

Speed | The rate of change of distance with time is known as speed. |

Velocity | The rate of change of displacement with respect to time is known as velocity. |

Acceleration | The rate of change of velocity with respect to time is known as acceleration. |

Uniform Motion | A body is said to be in uniform motion if it travels equal distance in equal intervals of time. It means the velocity remains constant during the motion. |

Non-Uniform Motion | A body is said to be in non-uniform motion if it travels unequal distance in equal intervals of time. |

__DISTANCE AND DISPLACEMENT__

__DISTANCE AND DISPLACEMENT__

**What is Distance?**

Let a particle has a starting point at ‘A=2m’ and later it comes to final point ‘B=4m’ after turning round through ‘C=6m’.

The distance travelled here = AC + BC = 4 + 2 = 6m

**“Distance is the total path length of the travelled by the particle in a given time interval.”**

- Path length of a body is a positive scalar quantity which doesn’t decrease with time and can never be zero for any moving body.
- Magnitude of distance is greater than r equal to the magnitude of displacement.

**What is Displacement?**

It is the * vector joining the intial and final position* of an object during a

*. The change in position of moving object is known as*

**time interval****displacement.**

- For a straight line motion, if a particle goes from A to B then,
- s = displacement = AB, the vector has only x component.

Hence its equal to the difference in X coordinates

** S= X _{b} – X_{a} = ∆X **

- If a particle goes from A to B along a curve in some time duration and if O is the origin then,

OA = initial position vector =r _{i}

OB= final position vector =r _{f}

AB= displacement of vector = OB – OA

s **=** r** _{f} –** r

_{i} ** |S|= (|r _{f}^{2}| – |r _{i}^{2}|)^{ 1/2}**

*Distance vs Displacement*

Distance | Displacement |

is the total path length of the travelled by the particle in a given time interval.Distance | It is the vector joining the intial and final position of an object during a time interval. The change in position of moving object is known as . displacement |

It’s a .positive scalar quantity | It’s a vector quantity. |

It has only magnitude. | It has .both magnitude and direction |

It during motion. In case of circular motion if a body completes 2 π rotation then the distance is 2 πrcannot be zero | It during motion. In case of a circular motion , if anybody completes 2 π rotation the displacement is 0can be zero |

It is always to the magnitude of displacement.greater than or equal | It’s the magnitude of distance.less than or equal to |

__AVERAGE AND INSTANTANEOUS VELOCITY/SPEED__

__AVERAGE AND INSTANTANEOUS VELOCITY/SPEED__

**AVERAGE SPEED:**It tells us how fast a particle moves in a particular interval.

It is a scalar quantity and is defined over an interval as Average Speed = Total Distance Covered/ Total Time Interval

Average speed has unit (m s^{-1}) and the magnitude of average speed is always greater than or equal to magnitude of average velocity.

**AVERAGE VELOCITY:**

*The average velocity is the vector in the direction of displacement. It depends only on the net displacement and time interval and not on the journey*.

**INSTANTANEOUS VELOCITY: **The velocity at an instant is defined as the limit of the average velocity as the time interval ∆t becomes infinitesimally small. In other words ,

*Velocity**at any instant is***equal**to the**slope of tangent**of**displacement time graph**.*It is the***average velocity**for**infinitely small time interval.***Thus instantaneous velocity =***tan***θ =**Magnitude of velocity in rectilinear motion at any time gives us speed of the particle.**The SI unit of Velocity is***m/s.**

****Note that for uniform motion, velocity is the same as the average velocity at all instants. Instantaneous speed or simply speed is the magnitude of velocity. Average speed over a finite interval of time is greater or equal to the magnitude of the average velocity, instantaneous speed at an instant is equal to the magnitude of the instantaneous velocity at that instant.*

__AVERAGE AND INSTANTANEOUS ACCELERATION__

__AVERAGE AND INSTANTANEOUS ACCELERATION__

**Average Acceleration***: The rate of change of velocity for a moving particle with respect to time is known as average acceleration.*

**Instantaneous Acceleration: ***The rate of change of velocity for a moving particle at any particular instant is known as instantaneous acceleration. Mathematically its expressed as ,*

*It is the***average acceleration**for**infinitely small time interval.***Acceleration**at any instant is***equal**to the**slope of tangent**of**velocity time graph**.*Thus instantaneous acceleration =***tan***θ**=*{\frac {dv} {dt}}*The SI unit of acceleration is***m/s**^{2}.

**If v = f(x) then** , ** a = {\frac {dv} {dt}} = {\frac {dv} {dx}}{\frac {dx} {dt}} = v{\frac {dv} {dx}} **

***When displacement is given and we need to calculate a, then we use**

a = {\frac {dv} {dt}} = {\frac {{d}^{2}x} {{dt}^{2}}}

__KINEMATICAL EQUATION FOR UNIFORMLY ACCELERATED MOTION__

__KINEMATICAL EQUATION FOR UNIFORMLY ACCELERATED MOTION__

For uniformly accelerated motion, we can derive some simple equations that relate **displacement** **(x), time taken (t), initial velocity (v _{0}), final velocity (v) and acceleration (a).**

Relation between final and initial velocities v and v_{0} of an object moving with uniform acceleration a can be derived from the relation:

a = {\frac {dv} {dt}}

*Integrating both sides we get:*

**v = v _{0} + at **

******This is called first equation of motion*

- The area under the curve represents the displacement over a given time interval.

** **This ** 2^{nd} equation of motion** is graphically represented in Fig. 3.12.

The **area** under this curve is: Area between instants 0 and t = Area of **triangle ABC** + **Area of rectangle OACD**

X= ½(v-v_{0}) t + v_{0}t

But v-v_{0 }= at Thus

**x= v _{0}t + ½* (a)* (t^{2})**

**For constant acceleration only**

**s= v _{0}t + ½* (a)* (t^{2})**

v_{avg }= s/t = (v_{0}t + ½* (a)* (t^{2}))/t

v_{avg }= u + ½ at = ½(2u+at) = ½(u + u + at)

** v _{avg}= ½(u+v)**

Where u = initial velocity

And v= final velocity

x= vt = ½*(v + v_{0})*(v-v_{0}) = (v^{2}-v_{0}^{2}) / 2a

**v ^{2}-v^{2}_{0}=2ax**

This is **3 ^{rd} equation** of motion.

All the above sets of equations are obtained by assuming that at t= 0 the position of the particle is zero. In case there is an x

_{0}displacement then the equations of motions are given by:

**v = v _{0} + at**

**x = x _{0 }+ v_{0}t + ½* (a)* (t^{2})**

**v ^{2}-v^{2}_{0}^{ }=2a(x-x_{0})**

__FREELY FALLING BODIES__

__FREELY FALLING BODIES__

When a body is dropped from some height which is much less than the radius of the earth, it falls freely under gravity with ** constant acceleration g** = 9.8 m/s

^{2}provided the air resistance is negligible.

The same set of kinematical equations are replaced with **a = g** and direction of y axis is chosen conveniently.

When*Case-1:*for*y axis is chosen positive*then we take*vertically downward motion***“g”**assince the direction of g is always downwards towards earth.*positive*

** v = v _{0} + gt **

** h = v _{0}t + ½* (g)* (t^{2})**

** v ^{2}-v^{2}_{0}= 2g(h)**

**where h is the vertical displacement**

** Case – 2:** When

**is taken**

*+ve y axis***for**

*+ve***then the equations of motions are :**

*vertically upward motion*The equations of motions are:

** v = v _{0} – gt**

** h = v _{0}t – ½* (g)* (t^{2}) **

**v ^{2}-v^{2}_{0}= -2g (h) **

** Note:**

*.*

**Choosing**of**sign conventions**is**necessary**. All the**vector**quantities should be assigned the**same convention**and the**convention**should be kept**fixed**throughout the**problem**. For example for case – 1 the displacement, velocity and g has a direction downward n we took downward as positive and rewrote the equations**Calculating the Displacement during n ^{th }second**

Displacement of n second- Displacement of (n-1)s

**= **un + ½(a)(n^{2}) – { u(n-1) + ½(a)(n-1)^{2}}

**=**u(n-n+1) + ½(a){n^{2} –( n-1)^{2}}

**s _{n}= u + ½(a)(2n-1)**

**Calculation of stopping distance:**

When brakes are applied to a moving vehicle, the distance it travels before stopping is called **stopping distance**. It is an important factor for road safety and depends on the initial velocity (v_{0}) and the braking capacity, or deceleration, –a is caused by the braking.

Let the distance travelled by the vehicle before it stops be ds. Then, using equation of motion v^{2} = v_{0}^{2} + 2 ax, and noting that v = 0, we have the stopping distance

**d _{s} = -v^{2} / 2a**

### POINTS TO REMEMBER WHILE PLOTTING GRAPHS

The theory of graphs can be generalised and summarised in following six points. :

- A
represents*linear equation*e.g. y = 4x. y= kx represents a straight line passing through origin in x- y graph.*straight line* represents*x= k/y*in x-y graph.*a rectangular hyperbola*- A
represents a*quadratic equation*in x- y graph .*parabola* , then value of z can be obtained by the*If z= dy/dx**slope*at that point.*of the graph*- If
, then the value of z between x*z = xy*_{1}and x_{2}between y_{1 }and y_{2 }can be obtained by thebetween x*area of graph*_{1}and x_{2}and y_{1}and y_{2 }.

**Important points**

*Slopes of v-t or s-t graph can never be***infinite**at any point, because infinite slope of v-t graph means infinite acceleration. Similarly, infinite slope of s-t graph means infinite velocity. Hence these graphs are not possible.*At one time only***two values of velocity**or displacement are not possible.*Different values of displacements in s-t graph corresponding to given v-t graph can be calculated just by calculating the area under v-t graphs.*

__MOTION WITH NON-UNIFORM ACCELERATION__

__MOTION WITH NON-UNIFORM ACCELERATION__

*Acceleration depends on time t*

Steps to find v (t) from a (t) By definition we have

a = {\frac {dv} {dt}}

Now integrating both sides,

Where v_{0 }= Initial velocity at time t=0

*Steps to find x(t) from v(t)*

To get x (t), we put v (t) = dx/dt

dx= v (t)dt

Integrating both sides,

Where x (0) = Position at t= 0

__Relative Velocity__

__Relative Velocity__

The word ** ‘relative’** means in relation or in proportion to something else.

Relative motion is the motion as observed from or referred to some system constituting a frame of reference.

The relative velocity of A with respect to B (written as ū_{AB }) means the velocity of A as seen from B . its represented as :

**ū _{AB} = ū_{A}– ū_{B}**

Similarly, relative acceleration of A with respect to B is

**ā _{AB }= ā_{A}– ā_{B}**

**For the further understanding of concepts related to equations of motion, graphical representation of motion in a straight line, relative velocity and freely falling bodies please refer to the PDF above in which these concepts are discussed in detail.**

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