Tiêu đề 3. One Dimensional Motion Theory.pdf ✅

82 EH EduHulk Download Free Study Material from EduHulk.com • Kinematics, Frame of Reference • Rest, Motion and Types of Motion • Position Vector, Displacement Vector and Distance • Average Velocity, Instantaneous Velocity, Average • Average Acceleration and Instantaneous Acceleration Speed and Instantaneous Speed • Motion and Graph • Elementary concepts of differentiation and integration • Motion Under Gravity for describing motion • Rain and River Problem • Uniform Accelerated Motion • Relative Motion Contents... ONE - DIMENSIONAL MOTION F Kinematics : Study of motion of objects without taking into account the factor which causes the motion (i.e. nature of force). F Motion & Rest : • If a body changes its position with time with respect to frame of reference, it is said to be moving with respect to that frame of reference else it is at rest. Motion and rest is always relative to the observer. • Motion and rest are a combined property of the object under study and the observer. There is no meaning of rest or motion without the observer or frame of reference. • If a body changs its position with time w.r.t. frame of reference, it is said to be in Motion. • The coordinate (x, y, z) of an object describe the position of the object w.r.t. this coordinate system. This coordinate system along with a clock (to measure the time) constitutes Frame of Reference. • If all the three coordinates of the particle remain unchanged as time passes, it means the particle is at rest w.r.t. this frame. The reference frame is chosen according to problems. If frame is not mentioned, then ground is taken as reference frame. à Note : In universe, absolute rest and absolute motion of particle is not possible F Types of motion : 1) Translatory motion :- When a body moves in such a way that the linear distance covered by each particle of the body is same during motion, the motion is said to be translatory.
83 EH EduHulk Download Free Study Material from EduHulk.com 2) Rotatory motion :- When a body moves in such a way that every moving particle of body describes the same angular displacement about axis of rotation, the motion is called rotatory motion. 3) Oscillatory Motion :- The motion in which an object repeats its motion about a fixed point back and forth is called oscillatory motion. Such a motion always takes place within well-defined boundaries, which are called extreme points. à Note : (i) Motion in 1-D :- If particle moves along straight line then this motion is in 1 -D. (ii) Motion in 2-D :- If particle move in a plane then this motion is in 2-D. Example : Projectile motion, circular motion. (iii) Motion in 3-D :- If particle moves in space then this motion is in 3-D. Example : Motion of kite. ________________________________________________________________________ Parameters of Motion F Position vector : • A vector which represents the position of a point about reference point is called position vector. For point (x, y, z) , position vector with respect to origin will be  r xi yj zk       . • Direction of position vector is from origin to that point P. F Displacement vector : • It is a vector joining the initial position of the particle to its final position. Mathematically it is equal to the change in position vector. • Its magnitude is equal to the shortest distance between the initial and final positions.   r is the displacement vector from point A to B.     r r r   B A F Distance : • Total length of path covered by the particle, during motion is known as distance. Let a body moves from A to B via C. The length of path ACB is called the distance travelled by the body. But overall, body is displaced from A to B. A vector from A to B, i.e. AB  is its displacement vector.
84 EH EduHulk Download Free Study Material from EduHulk.com à Note : (i) Distance is a scalar quantity and displacement is a vector quantity. (ii) Displacement can be negative, zero or positive, but distance is never negative. (iii) Distance does not decrease with time and is never zero for a moving body. (iv) In given time interval, |Displacement|  distance (v) Distance depends on path while displacement is independent of path. It depends upon initial and final position. (vi) If object move along a straight line without change in direction then distance and magnidue of displacement will be same. _______________________________________________________________________ Solved Examples... Example 1 : A particle moves from a point A(–1, 2, 3) m to a point B(2, –1, –2) m. Find the displacement of the particle. Solution : Position vector of A and B are  r i j k A        2 3 ;  r i j k B    2 2    Displacement of particle from A to B    S r r i j k      B A 3 3 5     | |  S  43 m Example 2 : A particle is moving in a circle of radius R. What is its distance and magnitude of displacement when it covers. (i) half revolution (ii) one revolution Solution : (i) Distance = 2 2 R = R Displacement Distance Displacement = 2R (ii) Distance = 2R Displacement = 0 [Particle returns to initial position] Example 3 : A displacement vector has x-component 5 m and makes an angle 60° with x-axis then find displacement vector? where  i and  j are unit vectors in x and y directions, respectively. Solution : Let displacement is  S then it’s x and y component will be given as Sx= 5 = Scos60° S = 10 m Sy = Ssin60° = 5 3 m  S i j   5 5 3   m _______________________________________________________________________ Try yourself... Question. 1 A particle initially at point A(2, 4, 6) m moves finally to the point B(3, 2, –3)m. Write the initial position vector, final position vector, and displacement vector of the particle. Question. 2 A particle has the following displacements in succession: (i) 12 m towards east, (ii) 5 m north, and (iii) 6 m vertically upwards. Find the magnitude of the resultant displacement. Answer : 1.  r i j k m A    2 4 6    ,  r i j k m B    3 2 3    ,  d i j k m       2 9 , 2. 205 m
85 EH EduHulk Download Free Study Material from EduHulk.com F Velocity : 1) Average Velocity :- It is the ratio of total displacement to the total time interval.      v r t r r t avg f i       Direction of average velocity is along the displacement vector. à Note : If a body is moving with uniform acceleration then average velocity is given by -  v final velocity Initial velocity avg   2     v v v avg i f   2 2) Velocity or Instantaneous velocity :- The rate of change of position i.e. rate of displacement with time is called velocity.  v  lim   t  r t dr  dt  0   Direction of instantaneous velocity is along to the tangent to the path at given instant. F Speed : 1) Average speed :- It is the ratio of total distance travelled by the particle to the total time taken. v avg = Total dis ce Total time d t tan   2) Speed or Instantaneous speed :- The rate of change of distance covered with respect to time is called speed. It is a scalar quantity. Magnitude of instantaneous velocity is equal to instantaneous speed. Dimension : [M0L 1T –1] Unit : meter/second (S.I.), cm/second (C.G.S.) à Note : (i) Speed is a scalar quantity while velocity is vector quantity. (ii) Speed can never be negative while velocity can be negative or positive or zero. (iii) In a given time interval, |Average velocity|  Average speed. (iv) The average speed of an object is greater than or equal to the magnitude of the average velocity over a given time interval. The two are equal only if the path length is equal to the magnitude of displacement. (v) Uniform velocity :- A particle is said to be moving with uniform velocity if magnitude and direction of velocity remains constant. If velocity is constant then particle moves along straight line and it covers equal displacement in equal time intervals. (vi) For uniform motion, velocity is the same as the average velocity at all instants.