REVSION PLUS – KINEMATICSLS-19/AS 18th,Feb,2010 7.30p.m : REVSION PLUS – KINEMATICSLS-19/AS 18th,Feb,2010 7.30p.m UNIFORM MOTION ACCELERATED MOTION MOTION IN A PLANE VECTORS
UNIFORM MOTION : UNIFORM MOTION REST.
An object is said to be at rest if it does not change its position with time w.r.t.its surroundings.
MOTION.
An object is said to be in motion if it changes its position with time, w.r.t.its surroundings.
UNIFORM MOTION : UNIFORM MOTION REST AND MOTION ARE RELATIVE.
It means an object in one situation is at rest in another situation the same object is in motion.
POINT MASS OBJECT.
An object can be considered as a point object if during motion in a given time, it covers a distance much greater than its own size.
UNIFORM MOTION : UNIFORM MOTION ONE DIMENSIONAL MOTION.
The motion of an object is said to be one dimensional motion if only one out of the three coordinates specifying the position of the object changes with respect to time.
In such a motion, an object moves along a straight line or on a well defined straight path.
UNIFORM MOTION : UNIFORM MOTION TWO DIMENSIONAL MOTION.
The motion of an object is said to be two dimensional motion if two out of the three coordinates specifying the position of the object change with respect to time. In such a motion, the object moves in a plane.
UNIFORM MOTION : UNIFORM MOTION THREE DIMENSIONAL MOTION.
The motion of an object is said to be three dimensional motion if all the three coordinates specifying the position of the object change with respect to time. In such a motion, the object moves in a space.
UNIFORM MOTION : UNIFORM MOTION TOTAL PATH LENGTH.
It is the total distance travelled by an object. It is defined as the length of the actual path transversed by an object during motion in a given interval of time.
Path length is a scalar quantity.
Its value can never be zero or negative, during the motion of an object.
UNIFORM MOTION : UNIFORM MOTION DISPLACEMENT.
The displacement of an object in a given interval of time is defined as the change in position of the object along a particular direction during that time and is given by the vector drawn from the initial position to final position.
The displacement of an object can be positive, zero or negative.
UNIFORM MOTION : UNIFORM MOTION The DISPLACEMENT of an object between two positions has a unique value, which is the shortest distance between them.
The magnitude of the displacement of an object in a given time interval can be equal or less than the actual distance travelled but never greater than total distance travelled.
UNIFORM MOTION : UNIFORM MOTION SPEED.
The speed of an object is defined as the ratio of path length covered and time taken during the change of position of the object in any direction i.e. speed=distance travelled/(time taken).
Speed is a scalar quantity.
It can be zero or positive but never negative.
UNIFORM MOTION : UNIFORM MOTION UNIFORM SPEED.
An object is said to be moving with a uniform speed, if it covers equal distances in equal intervals of time, howsoever small these intervals may be.
VARIABLE SPEED.
An object is said to be moving with a variable speed if it covers equal distances in unequal intervals of time,howsoever small these intervals may be.
UNIFORM MOTION : UNIFORM MOTION AVERAGE SPEED.
The average speed of an object for the given motion is defined As the ratio of the total distance travelled by the object to the total time taken
I.e. Average speed = total distance travelled/total time taken.
INSTANTANEOUS SPEED.
The speed of an object at a given instant of time is called its instantaneous speed. It is also defined as the first derivative of distance w.r.t.time.
UNIFORM MOTION : UNIFORM MOTION VELOCITY.
The velocity of an object is defined as the ratio of displacement of the object and time taken i.e.velocity = displacement/ time taken.
Velocity = speed + direction.
The velocity is a vector quantity.
The velocity of an object can be positive, zero or negative.
UNIFORM MOTION : UNIFORM MOTION UNIFORM VELOCITY
If an object undergoes equal displacements in equal intervals of time, it is said to be moving with a uniform velocity.
VARIABLE VELOCITY.
If an object undergoes unequal displacements in equal intervals of time or equal displacements in unequal intervals of time, it is said to be moving with a variable velocity.
UNIFORM MOTION : UNIFORM MOTION AVERAGE VELOCITY.
The average velocity of an object is equal to the ratio of the total displacement, to the total time interval for which the motion takes place. Call me at……………9814123832
Email ………………. hksidhuinstitute@gmail.com
UNIFORM MOTION : UNIFORM MOTION INSTANTANEOUS VELOCITY.
The velocity of an object at a given instant of time is called its instantaneous velocity.
When a body is moving with a uniform velocity, its instantaneous velocity = average velocity = uniform velocity.
Instantaneous velocity of an object is also defined as the first time derivative of displacement at the given instant.
UNIFORM MOTION : UNIFORM MOTION 19.Uniform motion in a straight line. An object is said to be in uniform motion if it undergoes equal displacements in equal intervals of time, however small these intervals may be.The position of an object in uniform motion in a straight line is given by are the displacements of an object at timings t = 0 and t = t. And is the uniform velocity of the object.
UNIFORM MOTION : UNIFORM MOTION Velocity-time graph of a uniform motion
in one dimension is a st. line parallel to time axis.
The area enclosed by this graph with time axis
measures the displacement of an object in a given interval of time.
Position-time graph of a uniform motion
in one dimension is a straight line inclined to time axis.
The slope of the position-time graph with time axis
tells the velocity of the object.
UNIFORM MOTION : UNIFORM MOTION RELATIVE VELOCITY.
The relative velocity of one object w.r.t. another is the velocity with which one object moves w.r.t. another object.
If are the velocity of two objects A and B, and ? is the angle between them, then relative velocity of object A w.r.t.B is given by
UNIFORM MOTION : UNIFORM MOTION RELATIVE VELOCITY. .
UNIFORMLY ACCELERATED MOTION : UNIFORMLY ACCELERATED MOTION Acceleration.
The acceleration of an object is defined as the ratio of change of velocity of the object, and time taken i.e. Acceleration = change in velocity/time taken.
Acceleration is a vector quantity. Acceleration is positive, if the velocity is increasing and is negative if velocity is decreasing.
The negative acceleration is called retardation or deceleration.
UNIFORMLY ACCELERATED MOTION : UNIFORMLY ACCELERATED MOTION UNIFORM ACCELERATION.
If the velocity of an object changes by equal amounts is equal intervals of time, it is said to be moving with a uniform acceleration.
VARIABLE ACCELERATION
If the velocity of an object changes by unequal amounts in equal intervals of time, it is said to be moving with a variable acceleration.
UNIFORMLY ACCELERATED MOTION : UNIFORMLY ACCELERATED MOTION AVERAGE ACCELERATION.
The average acceleration of an object for a given motion is defined as the ratio of the total change in velocity of the object during motion to the total time taken
i.e. average acceleration = total change in velocity/total time taken.
UNIFORMLY ACCELERATED MOTION : UNIFORMLY ACCELERATED MOTION INSTANTANEOUS ACCELERATION.
The acceleration of an object at a given instant or at a given point of motion is called its instantaneous acceleration.
It is defined as the first time derivative of velocity at a given instant or it is also equal to the second time derivative of the position of the object at a given instant i.e.
Instantaneous acceleration,
Formulae for uniformly accelerated motion along a straight line. : Formulae for uniformly accelerated motion along a straight line.
UNIFORMLY ACCELERATED MOTION : UNIFORMLY ACCELERATED MOTION VELOCITY-TIME GRAPH OF A UNIFORMLY ACCELERATED MOTION
is a st. line inclined to time axis.
THE SLOPE OF THIS ST. LINE 0F V-T GRAPH
tells the acceleration of the objected
AREA ENCLOSED BY THE ST. LINE WITH TIME AXIS
tells the distance travelled by object in a given time interval.
VECTORS : VECTORS SCALARS.
There are those quantities which have only magnitudes but no direction. For example, mass, length, time, speed,work,temperature etc.
VECTORS.
These are those quantities which have magnitude as well as direction. For example,displacement,velocity,acceleration,force, momentum etc.
VECTORS : VECTORS UNIT VECTOR.
A unit vector of a given vector is a vector of unit magnitude and has the same direction as that of the given vector. A unit vector of is written as
A unit vector is unitless and dimensioniess vector and respresents direction only.
VECTORS : VECTORS EQUAL VECTORS.
The two vectors are said to be equal if they have equal magnitude and same direction.
NEGATIVE VECTORS.
A negative vectors of a given vector is a vector of same magnitude but acting in a direction opposite to that of the given vector.
The negative vector of is represented by -
VECTORS : VECTORS COLLINEAR VECTORS.
These are those vectors which are acting along parallel straight lines.
COPLANAR VECTORS.
These are those vectors which are acting in the same plane.
LOCALISED AND NON-LOCALISED VECTORS.
A vector whose initial point is fixed is called a localised vector and whose initial point is not fixed is called non-localised vectors.
VECTORS : VECTORS Multiplication of a vector by a real number. When a vector is multiplied by a real number n, it becomes another vector
n Its magnitude becomes n times the magnitude of .
Its direction is same or opposite as that of , according as n is positive or negative real number. The unit of n is the same as that of .
VECTORS : VECTORS Multiplication of a vector by a scalar. When a vector is multiplied by a scalar S, it becomes a vector S , whose magnitude is S times the magnitude of and it acts along the direction of . The unit of S is different from the unit of vector .
VECTORS : VECTORS RESULTANT VECTOR.
The resultant vector of two or more vectors is defined as that single vector which produces the same effect as is produced by individual vectors together. The resultant vector can be obtained from the rules for geometric addition of vectors.
VECTORS : VECTORS (I) For the addition of two vectors, represent these two vectors by arrowed lines using the same suitable scale.
Displace the second vector such that its tail coincides with the head of first vector. Then the single vector, drawn from the tail of the first vector to the head of the second vector represents the resultant vector.
VECTORS : VECTORS (ii) For the addition of three or more vectors, represent these vectors by arrowed lines using the same suitable scale.
Displace these vectors such that the head of the first vector coincides with the tail of second vector and whose head coincides with the tall of third vector and so on,
then the single vector drawn from the tail of the first vector to head of last vector represents resultant vector.
VECTORS : VECTORS Analytical method of vector addition.
(i) TRIANGLE LAW OF VECTORS.
It states that if two vectors acting on a particle at the same time are represented in magnitude and direction by the two sides of a triangle taken in one order, their resultant vector is represented in magnitude and direction by the third side of ,the triangle taken in the opposite order:
VECTORS : VECTORS If is the resultant of ,Fig,then
VECTORS : VECTORS PARALLELOGRAM LAW OF VECTORS.
It states that if two vectors acting on a particle at the same time be represented in magnitude and direction by the two adjacent sides of a parallelogram drawn from a point, their resultant vector is represented in magnitude and direction by the diagonal of the parallelogram drawn from the same point.
VECTORS : VECTORS If is the resultant of ,Fig,then
Note .The magnitude of the resultant vector is maximum if the two vector are acting in the same direction and is minimum if the two vectors are acting in the opposite direction.
VECTORS : VECTORS POLYGON LAW OF VECTORS.
It states that if number of vectors acting on a particle at a time are represented in magnitude and direction by the various sides of an open polygon taken in same order, their resultant vector is represented in magnitude and direction by the closing side of the polygon taken in opposite order.
In fact polygon law of vectors is the outcome of triangle law of vectors.
VECTORS : VECTORS Important points of vector addition.
(i) Vector of same nature alone can be added I.e. a force vector can not be added to velocity vector but can be added to force vector only.
(ii) Vector addition is commutative i.e.
Vector addition is associative i.e.
VECTORS : VECTORS Subtraction of vectors.Subtraction of a vector from a vector is defined as the addition of vector - (negative of vector ) to vector .Thus
If ? is the angle between
VECTORS : VECTORS The vector subtraction does not obey, commutative law and associative law.
VECTORS : VECTORS Zero vector. It is that vector which has zero magnitude and an arbitrary direction. A zero vector is represented by . It is also called a null vector. Call me at……………9814123832
Email ………………. hksidhuinstitute@gmail.com
VECTORS : VECTORS Rectangular components of a vector in a plane.
When a vector is splitted into two component vectors at right angles to each other, the component vectors are called rectangular components of a vector. If makes an angle ? with x-axis and
be the rectangular components of along x-axis respectively,
VECTORS : VECTORS
VECTORS : VECTORS Addition of vector in rectangular coordinates.
MOTION IN A PLANE : MOTION IN A PLANE PROJECTILE.
Projectile is the name given to a body which is thrown with some initial velocity with the horizontal direction and then it is allowed to move under the effect of gravity alone, without being propelled by any engine or fuel.
A projectile during its flight must posses two component velocities(i) in the horizontal direction and (ii) in the vertical direction
MOTION IN A PLANE : MOTION IN A PLANE If a projectile given angular projection of angle ? with the horizontal direction with velocity u, then its
MOTION IN A PLANE : MOTION IN A PLANE There are two angles of projections(? and ?)for which the horizontal range will be the same provided ? + ?=90o. Call me at……………9814123832
Email ………………. hksidhuinstitute@gmail.com
MOTION IN A PLANE : MOTION IN A PLANE Angular displacement(?).
The angular displacement of an object moving around a circular path is defined as the angle traced out by the radius vector at the axis of the circular path in the given time.
Angular displacement is a vector quantity provided angle ? is small. It is expressed in radians.
MOTION IN A PLANE : MOTION IN A PLANE Angular velocity (?).
It is defined as the time rate of charge of angular displacement of the object i.e. ? = d?/dt. Its S.I.unit is rad/s.
Uniform circular motion. When a point object is moving on a circular path with a constant speed, then the motion of the object is said to be a uniform circular motion.
MOTION IN A PLANE : MOTION IN A PLANE CENTRIPETAL ACCELERATION. It is defined as the acceleration acting on the object undergoing uniform circular motion. It always acts on the object along the radius towards the centre of the circular path. The magnitude of centripetal acceleration is
Slide 54 : The distance of a particle moving along a straight line from another stationary point on same line is given by l =2+4t+5t2. With l in metre and t in second. The distance travelled by the body between 4th and 6th seconds will be…. Let’s solve the next problem… Sol: Given, l = 2 + 4t + 5t2 Distance travelled by the body between 4th & 6th second = 108 m = S(6) – S(4) = [2 + 4?? 6 + 5 ? 62] – [2 + 4 ? 4 + 5 ? 42] = [2 + 24 + 180] – [2 + 16 + 80] = 206 – 98 = 108 m Distance travelled by the body between 4th & 6th second
. : . . h v/3 v/2 A stone projected so as to reach a height h passes P and Q with velocities v/2 and v/3. The distance between the points is _______ h , where v is the initial velocity with which the body is thrown . CASE–1 CASE–2 v u 2 v1 v2 h S2 = X =? a1 = a2 Applying for both the cases we get, u = v S1 = u 1 = = v/2 = 0 = v/3 –g = – g Sol: (AB) (PQ) Formula: (2) (1) X=?
Slide 56 :