physics-kinematics

TERM- I2013-14

By, By, Akshatha NayakAkshatha Nayak

XI-BXI-BRoll no. 23Roll no. 23

KV AFS, BareillyKV AFS, Bareilly

KINEMATICSKINEMATICS►It is the branch of mechanics that is

concerned with the motion of objects.►Motion of objects can be described in 3 different ways:-MOTION

1-DIMENSIONAL

2-DIMENSIONAL

3-DIMENSIONAL

Only 1 coordinate is required to specify the position of the object i.e. x coordinate.

Two coordinates are required to specify its position i.e. x and y coordinate.

Three coordinates are required for specifying its position i.e. x, y and z coordinate.

Scalars VS VectorsScalars VS Vectors► A scalar is a quantity

that has only magnitude and no direction.

► It is always positive.

► Examples of scalar quantities are- distance, speed, mass and temperature

► A vector is a quantity that has both magnitude as well as direction.

► It can be positive , negative or zero.

► Examples of vector quantities are -displacement, velocity and acceleration.

Distance VS DisplacementDistance VS Displacement► Distance is the total

path length covered by an object.

► It is a Scalar Quantity.► In the given fig. the

curved path AOCDB shows the distance traveled by the object.

► Displacement is the shortest distance between the initial and the final position.

► It is a Vector Quantity.► In the given fig. the

straight line AB shows displacement.start

stop

Speed VS VelocitySpeed VS Velocity► Speed is a scalar

quantity i.e. how fast something is moving regardless of its direction.

► Ex: v = 20 mph

► SI Unit is m/s.

► The symbol for speed is v.

► Velocity is a combination of speed and direction. It is a vector quantity.

► Ex: v = 20 mph at 15 south of west

► SI Unit is m/s.

► The symbol for velocity is type written in bold: v or hand written with an arrow: v.

AVERAGE SPEED AND AVERAGE AVERAGE SPEED AND AVERAGE VELOCITYVELOCITY

► Average speed is defined as the total path length covered by an object divided by the total time interval.

► Average speed = Total Path length Total time taken

► Average velocity is defined as the rate of change of displacement of an object with respect to time.

► Average velocity = Displacement Time interval

Instantaneous VelocityInstantaneous Velocity ►Instantaneous Velocity-

Instantaneous Velocity or the velocity of an object at an instant of time is defined as the limit of the average velocity as time interval ∆t 0 approaches to zero. If velocity is If velocity is constant, then constant, then

VVinstinst = V = Vavav

► If velocity is not constant then If velocity is not constant then VVinstinst = lim ∆x/ ∆t = dx/dt = lim ∆x/ ∆t = dx/dt

∆ ∆t t 0 0

ACCELARATIONACCELARATION AVERAGE ACCELARATION- The average

acceleration a over a period of time is defined as the rate of change of the velocity of an object with respect to the time interval.

a = Vf – Vi / t f – t i = ∆V/∆t INSTANTANEOUS ACCELARATION- Instantaneous

acceleration or the acceleration of an object at an instant of time is defined as the average acceleration as time interval approaches to zero.

aainstinst = lim = lim ∆V/ ∆t∆V/ ∆t ∆∆tt 0 0

= dv/dt

Velocity & Acceleration Sign Velocity & Acceleration Sign ChartChart

V E L O C I T YV E L O C I T Y AACCCCEELLEERRAATTIIOONN

++ --++ Moving forward;Moving forward;

Speeding upSpeeding up

Moving Moving backward;backward;

Slowing downSlowing down -- Moving forward;Moving forward;Slowing downSlowing down

Moving Moving backward;backward;

Speeding upSpeeding up

Acceleration due to Acceleration due to GravityGravity

9.8 m/s2

FREE FALL- In the absence of the air resistance all bodies fall with the sameacceleration towards earth from a small height. This is called free fall. The acceleration with which a body falls is called gravitational acceleration (g). aa = - = -gg = -9.8 m/s = -9.8 m/s22

Velocity decreases by 9.8 m/s each second, meaning velocity is becoming less positive or more negative. Less positive means slowing down while going up. More negative means speeding up while going down.

DERIVATION OF DERIVATION OF KINEMATIC KINEMATIC

EQUATIONS OF EQUATIONS OF MOTION BY MOTION BY GRAPHICAL GRAPHICAL

METHODMETHOD

FIRST EQUATION OF MOTION Consider an object moving with a

uniform velocity u in a straight line. Let it be given a uniform acceleration a at time t = 0 when its initial velocity is u. As a result of the acceleration, its velocity increases to v (final velocity) in time t and S is the distance covered by the object in time t.

The figure shows the velocity-time graph of the motion of the object.

Slope of the v - t graph gives the acceleration of the moving object.

Thus, acceleration = slope = AB = BC/AC = (v-u)/t v - u = at v = u + at

SECOND EQUATION OF MOTION Let u be the initial velocity of an object

and 'a ' the acceleration produced in the body. The distance travelled S in time t is given by the area enclosed by the velocity-time graph for the time interval 0 to t.

Distance travelled S = area of the

trapezium ABDO = area of rectangle ACDO + area

of DABC = OD X OA + ½(AC X BC) = t(u) + ½ t(v-u) From 1st eqn of motion, v-u =at . So, s = ut + ½ t (at) s = ut+ ½at2

S = ut + ½at2

THIRD EQUATION OF MOTIONLet 'u ' be the initial velocity of an

object and a be the acceleration produced in the body. The distance travelled 'S ' in time 't ' is given by the area enclosed by the v - t graph.

S = area of the trapezium OABD. = ½(OA + BD) OD = ½(u+v) t from 1st eqn of motion, t= (v-u)/a S = ½(u+v) (v-u)/a S = ½(v2-u2)/a v2-u2 = 2aS

DERIVATION OF DERIVATION OF KINEMATIC KINEMATIC

EQUATIONS OF EQUATIONS OF MOTION BY MOTION BY CALCULUS CALCULUS METHODMETHOD

FIRST EQUATION OF MOTION

Velocity−Time Relation

Acceleration, a = dv/dt or dv = adt Integrating the above,

v − uv − u = = atat or or v = u + at v = u + at

u → Initial velocityv → Final velocitya → Acceleration

t → Timex0→ Initial position

x→ Position after time t

SECOND EQUATION OF MOTION

Displacement−Time RelationInstantaneous velocity, v= dx/dt or vdt = dx From 1st eqn of motion (u+at) dt = dxIntegrating both sides, we get

x-x0 = u(t-0) +½at2

S= ut + ½at2




THIRD EQUATION OF MOTION

Velocity−Displacement Relation

Acceleration, a = dv/dt a = (dv/dx) (dx/dt) a = v dv/dxadx = v dvIntegrating both sides, we get,

a(x-x0) = v2/2 – u2/2

aS = (v2– u2)/2 v2– u2 = 2aS




RELATIVE VELOCITYRELATIVE VELOCITY The relative

velocity or the velocity of of a body A with respect to another body B is the time rate at which A changes its position with respect to B.

Case 1: Both bodies move with the same velocity in

same direction. If A and B have equal

velocity ,moving in the same direction, then ,

VAB=VA-VB

= VA-VA

VAB= 0

Also, VAB =- VBA

Case 2: Both bodies move in the same direction with

different velocities.

If A and B are moving in the same direction, with different velocities, then the relative velocity of A with respect to B is VAB=VA-VB.

Case 3: The bodies move in opposite directions.

If A and B are moving in the opposite directions, then the relative velocity of A with respect to B is VAB=VA -(-VB) =VA +VB

POSITION - TIME POSITION - TIME GRAPHS GRAPHS

DEPICTING DEPICTING DIFFERENT TYPES DIFFERENT TYPES

OF MOTIONOF MOTION

OBJECT MOVING WITH UNIFORM VELOCITY

OBJECT MOVING WITH ACCELARATED MOTION

STATIONARY OBJECT

OBJECT MOVING WITH RETARDED MOTION

VELOCITY - TIME VELOCITY - TIME GRAPHS GRAPHS


OF MOTIONOF MOTION

OBJECT MOVING WITH POSITIVE OBJECT MOVING WITH POSITIVE VELOCITYVELOCITY

OBJECT MOVING WITH OBJECT MOVING WITH NEGATIVE VELOCITYNEGATIVE VELOCITY

ACCELERATION - ACCELERATION - TIME GRAPHS TIME GRAPHS


OF MOTIONOF MOTION

STATIONARY OBJECT OBJECT MOVING WITH UNIFORM VELOCITY

ZERO ACCELERATIONZERO ACCELERATION

NON-ZERO ACCELERATIONNON-ZERO ACCELERATION

UNIFORMLY ACCELERATED MOTION

NON-UNIFORM ACCELERATION

BY,BY,AKSHATHA NAYAKAKSHATHA NAYAK

ROLL NO: 23ROLL NO: 23CLASS XI-BCLASS XI-B

KV AFS BAREILLYKV AFS BAREILLY

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