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  • KINEMATICS

    DHANALAKSHMI NAGAR

    NEAR ANNAMAIAH CIRCLE,

    TIRUPATI.

    PH NO. 9440025125

  • KINEMATICS

    www.physicsashok.in 1 KINEMATICS

    THEORY OF KINEMATICSKinematics : The study of the motion of an object without taking into consideration cause of its motion is called

    kinematics.

    NOTE : The word kinematics comes from the Greek word Kinema which means

    motion. The word dynamics comes from the Greek word dynamics which means power.

    BASIC DEFINITIONSDistance and Displacement

    Suppose an insect is at a point A(x1, y1, z1) at t = t1. It reaches at point B(x2, y2, z2) at t = t2 through path ACBwith respect to the frame shown in fig. The actual length of curved path ACB is the distance travelled by theinsect in time t = t2 t1.

    A B

    C

    X

    Y

    Z

    Ar Br

    If we connect point A (initial position) and point B (final position) by a straight line, then the length of straight lineAB gives the magnitude of displacement of insect in time interval t = t2 t1.The direction of displacement is directed from A to B through the straight line AB from the concept of vector,the position vector of A

    is A 1 1 1 r x i y j z k= + +

    and that of B is B 2 2 2 r x i y j z k= + +

    .According to addition law of vectors,

    A Br AB r+ =

    B AAB r r= -

    ( ) ( ) ( )2 1 2 1 2 1 AB x x i y y j z z k= - + - + -

    The magnitude of displacement is

    ( ) ( ) ( )2 2 22 1 2 1 2 1| AB | x x y y z z= - + - + -

    NOTE : Distance covered by the body is always equal to or greater

    than its magnitude of displacement.

    Example 1. A man walks 3m in east direction, then 4m in north direction. Find distance covered and thedisplacement covered by man.

    Sol. The distance covered by man is the length of path = 3m + 4m = 7m.Let the man starts from O and reaches finally at B (shown in figure).

    OB

    represents the displacement of man. From figure,

    ( ) ( )2 2| OB | OA AB= +

    ( ) ( )2 2| OB | 3m 4 m 5 m= + = W E

    N

    S

    O 3 m A

    B

    4 m

  • KINEMATICS

    www.physicsashok.in 2 KINEMATICS

    and4m 4

    tan3m 3

    q = =

    1 4tan

    3- q=

    The displacement is directed at an angle 14

    tan3

    - north of east.

    Average Speed and Average VelocitySuppose we wish to calculate the average speed and average velocity of the insect (in section (i)) betweent = t1 and t = t2. From the path (shown in fig.) we see that at t = t1, the position of the insect is A(x1, y1, z1) andat t = t2, the position of the insect is B(x2, y2, z2).

    A B

    C

    X

    Y

    Z

    Ar Br

    The average speed is defined as total distance travelled by a body in a particular time interval divided by thetime interval. Thus, the average speed of the insect is

    av2 1

    The length of curve ACBv

    t t=

    -The average velocity is defined as total displacement of the body in a particular time interval divided by the timeinterval.Thus, the average velocity of the insect in the time interval t2 t1 is

    av2 1

    ABv

    t t=

    -

    B Aav

    2 1

    r rv

    t t-

    =-

    ( ) ( ) ( )2 1 2 1 2 1av

    2 1

    x x i y y j z z kv

    t t- + - + -

    =-

    Example 2. In one second a particle goes from point A to point B moving in a semicircle (fig.).Find the magnitude of average velocity.

    Sol. avAB

    | v |t

    =D

    A

    B

    1 0m.

    av2.0

    | v | m / s1.0

    =

    av| v | 2 m / s=

    Example 3. A particle goes from A to B with a speed of 40 km/h and B to C with a speed of 60 km/h. If AB =6BC the average speed in km/h between A and C is _______

    Sol. AB = 40t1 ...(1)BC = 60t2 ...(2)

  • KINEMATICS

    www.physicsashok.in 3 KINEMATICS

    total distance travelledAverage speed =time taken

    av1 2

    AB BCVt t

    +=+

    From eqn. (1) and (2) A B C1 2

    av1 2

    40t 60tVt t

    +=+ ...(3)

    According to questionAB = 6BC40t1 = 6 60t2 From eqn (1) and (2)t1 = 9t2

    From eqn (3)2 2

    av2 2

    40 9t 60tV9t t

    from eqn (3)

    2av

    2

    420tV10t

    = avV 42 km / h=

    Instantaneous VelocityInstantaneous velocity is defined as the average velocity over smaller and smaller interval of time.Suppose position of a particle at t is r

    and at t + t is r r+ D . The average velocity of the particle for time

    interval t is avr

    vt

    D=

    D

    .

    From our definition of instantaneous velocity, t should be smaller and smaller. Thus, instantaneous velocity is

    t 0

    r drv lim

    t dtD D

    = =D

    Example 4. Let at any time t, the position vector of a particle is r x i y j z k= + +

    . Find the velocity of the particle.

    Sol. x-component of velocity, xdx

    vdt

    =

    y-component of velocity, ydy

    vdt

    =

    z-component of velocity, zdz

    vdt

    =

    Thus, velocity of particle

    x y 2 v v i v j v k= + +

    dx dy dz v i j kdt dt dt

    = + +

    Average and Instantaneous AccelerationIn general, when a body is moving, its velocity is not always the same. A body whose velocity is increasing issaid to be accelerated.Average acceleration is defined as change in velocity divided by the time interval.Let us consider the motion of a particle. Suppose that the particle has velocity 1v

    at t = t1 and at a later time

  • KINEMATICS

    www.physicsashok.in 4 KINEMATICS

    t = t2 it has velocity 2v

    . Thus, the average acceleration during time interval t = t2 t1 is

    2 1av

    2 1

    v v va

    t t t- D

    = =- D

    If the time interval approaches to zero, average acceleration is known as instantaneous acceleration.Mathematically,

    t 0

    v dva lim

    t dtD D

    = =D

    Example 5. The velocity of a point depends on time t, as v c t b= +

    where c

    and b

    are constant vectors.Find the instantaneous acceleration at any time t.

    Sol. Acceleration at any time t,

    ( )dv da c t bdt dt

    = = +

    a 0 b b= + =

    IMPORTANT FEATURES1. If a body is moving continuously in a given direction on a straight line, then the magnitude of displacement is

    equal to distance.2. Generally, the magnitude of displacement is less or equal to distance.3. Many paths are possible between two points. For different paths between two points, distances are different

    but magnitudes of displacement are same.4. The slope of distance-time graph is always greater or equal to zero.5. The slope of displacement-time graph may be negative.6. If a particle travels equal distances at speeds v1, v2, v3, ...... etc. respectively, then the average speed is

    harmonic mean of individual speeds.

    7. If a particle moves a distance at speed v1 and comes back with speed v2, then 1 2

    av1 2

    2v vv

    v v=

    +but avv 0=

    8. If a particle moves in two equal intervals of time at different speeds v1 and v2 respectively, then1 2

    avv v

    v2+

    =

    9. The average velocity between two points in a time interval can be obtained from a position versus time graphby calculating the slope of the straight line joining the co-ordinates of the two points.

    x2x1

    t1 t2(a)

    x2x1

    t1 t2(b)

    ( )x2 x1( )t2 t1

    BA

    The graph [shown in fig.], describes the motion of a particle moving along x-axis (along a straight line).Suppose we wish to calculate the average velocity between t = t1 and t = t2. the slope of chord AB [shown infig.(b)] gives the average velocity.

  • KINEMATICS

    www.physicsashok.in 5 KINEMATICS

    Mathematically, 2 1av2 1

    x xv tan

    t t-

    = q=-

    10. If a body moves with constant velocity, the instantaneous velocity is equal to average velocity. The instantaneousspeed is equal to modulus of instantaneous velocity.

    11. x-component of displacement is xx v dtD = y-component of displacement is yy v dtD = z-component of displacement is zz v dtD = Thus, displacement of particle is

    r x i y j z kD = D + D + D

    12. If particle moves on a straight line, (along x-axis), then dx

    v .dt

    =

    13. The area of velocity-time graph gives displacement.14. The area of speed-time graph gives distance.15. The slope of tangent at position-time graph at a particular instant gives instantaneous velocity at that instant.16. The slope of velocity-time graph gives acceleration.17. The area of acceleration-time graph in a particular time interval gives change in velocity in that time interval.

    ONE, TWO AND THREE DIMENSIONAL MOTIONOne Dimensional Motion

    If only one of the three co-ordinates is required to specify the position of an object in space changing w.r.t.time, then the motion of the object is called one dimensional motion. Motion of a particle in a straight line canbe described by only one component of its velocity and acceleration. For example, motion of a block in astraight line, motion of a train along a straight track, a man walking on a level and narrow road, an object fallingunder gravity, etc.

    Two Dimensional MotionIf two of the three co-ordinates are required to spacify the position of an object on space changing w.r.t. time,then the motion of the object is called two dimensional motion. The motion of a particle through its verticalplane at some angle with horizontal. ( 90) is an example of two dimensional (2D) motion.This is a projectile motion. Similarly, a circular m