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(2) FRAME OF REFERENCE (2) DISTANCE AND DISPLACEMENT + A rectangular coordinate system consisting of them | | * Distance: Actual path length in notion. During mation it mulivarly perpendicular axes, usually in the form of is never negelive or zare. a point of intersection of these thsee axes is called origin. | | * Displacement: The shertest path between initial and (0) final position. It can be positive, negative or rare. age ‘ lacement is a vecter quantity. It may or may = If a body changes its position on time pasoes wart. furan || PMP of reference, its motion is said to be in motion. 5 a ed ae 2 * Motion of objects along a straight line is called rectliliner x (m) x (m) motion. 40 z 30 20 10 y 9 tis) &(s) © 10 20 30 40 SO x Fig: Stationary object So ee \v = () veLocity (S) ACCELERATION time and in what direction. ; _ Ve-¥, he acceleration + Average velocity Vax = a Average Acceleration a, aa = me 3 0 : « Units are ms”! Instantanoous tanker! dv _ dv ; F . on eres Stas dt = dt Instantaneous velocity v= lim 7% = <= fo er > Fig: Negative , ne h acceleration ~ 7.7 al ge cela ag Uniform Acceleration a i ; i i velocity in Fig: Moviug with Fig: Moving with Equation of motion for 5 t positive velocity negative velocity wr op ae i % Non-Uniform Acceleration ozo Fig: Zero Acceleration changes with time —_—_— acceleration 0 t z KINEMATIC EQUATIONS (7) FoR MOTION WITH VARIABLE ACCELERATION * A mathematical equation to describe the motion of a oa OF a ae eS favs fade “ v-u+ fade body in one dimension. y dt For conformly accelerated motion A B B eve & => dv=vdt = fav - Jude => ee t ° adv na =» vdv=adx => Jvdv = fadx * v= utat (v-u) at * v= u'+ 2as U\ > [t (v?-u5) =a dx * ss = (us vt (1, se = zs = da = dt => =a. dt "se & (2n- 1) at? E t— - qe i aie ng J Je [a - a - Jade] E Je "v= Gein — at . dv * y = (use ¥, dt* ae ® FOR MOTION UNDER GRAVITY Fig: Variatien of Fig: Variation ef (9) Relative Velocity A mathematical treatment to describe velocity with time displacement with time The velocity with which an object moves the metion of & body in one dimension vy = | y y (m) with objects to another object is called under free fell 7. 0 0 relative velecity. * Vertically downmard metion | | = 2 3 4t(s) 2 4(s)) |, Va = Va- Va Fig: Position-time graphs of te ie pin -20 -30 *V, = Va = Va two objects with equel velociies. * 2 -30 > . Vv, = Va - Vv a (m) A * Vertically upwuad motion t(s) = oa 3 80 -40 -60 F vie B n= = a -70 Vea = Ya ~ Ye - y= ut ~ $gt? z ipod * Voe = (Va - Vo) v? = u* = 2gy -9.8 ms 2 ry =2-10123 4 5 6t(s) * Distance traveled during iii te. ———_—— — — NN NN of time by a body, falling boely from Fig: Position-time graphs of Fig: Position-time graphs of Fig: Paosition-time graphs of rest is in ratio 1:3:5:7:9:11... two objects moving in same two objects moving In two objects with velocities in (Gabilee's line) direction, showing the time opposite directions, showing opposite directions, showing the of meeting. the time of meeting. time of meeting. . 140 B 140 A % “ « : 60 20 20 -20 + + > t(s) A + “a0 t(s) 00 1234 5 6 400 123456 400 1 2 3 45% Motion in a Plane y @ SCALARS AND VECTORS * Scalar quantity: It has only magnitude with proper unit. All scalars are ruled out by ordinary algebra rules. * Vector quantity: It has both magnitude and direction with proper unit. They obey the triangle law or parallelogram a i + Parallelogram law of vector addition: For two | | @) RESOLUTION OF VECTORS co-initial vectors represented by two adjacent sides of a parallelogram, the resultant parallelogram + a > - passing through same point will be the resultant. = = OQ + QP 6B - --- <-=-5=--= A = Aa + pb A vector can be split into many vectors. ~ law of vector addition. P « Equality of vector: Two vectors A and B O A are said to be equal in magnitude and >a p nm direction only if they have same magnitude = A? + B? + 2AB cosé H and direction. Bsin@ ie) a an + Multiplication of vector by real numbers: If A+Bcos@ O ha Q a vector A is multiplied by real number, then Sulkin chee belie: nica bie fihall |AA| = |A| |A| as addition of a vector and negative of other vector. c i.e., magnitude will change and direction remains | A -B =A +(-B) = |A-B| = VA? +B?-2ABcos@ 5] MOTION IN A PLANE WITH same if A> 0, magnitude changes A times and CONSTANT ACCELERATION direction gets reversed if A < 0. B i = = = R V = Vo + at F = fy + Vot + Lat? f A . € 7 J F= hy + Vot + Lat ( > 4 ~ 2 ©) RECTANGULAR COMPONENTS @ MOTION IN A PLANE 1 Z ig: we. y “€ 4 , X = Xo + Voxt + > axt © A=A,+ A, F = xi + yj - A, = Acos@; A, = Asin@ ,ire * . 4 rF=x tt y's \ y |e |Al = VA; +A? s- th id y ieee SMa. eet n'(42) en? * 5 © @ RELATIVE VELOCITY IN 1 A, = (x’-x)t + (y’-y)j TWO DIMENSIONS * Resolution in three rectangular components Ar= r"-F' The velocity of object A relative to B A=AL+AlIt+Ak where Va and Ve are velocities . - : = 3 x')i “hy -9 \y in the same frame “4 2 3 2 : |A| = JA? + A2 +A: ) irection of ¥ Instanta ity, > 2 a oe m Direction of V nstantaneous velocity Vag = Va - Vp A _ Cs dx A Zz r x 2} én aA - 2 a> dt = dt »\ > '* v xt tv Similarly, Vea = Vp = Va The direction of velocity at any point on path is Z = * tangent to path and in the direction of motion. a | Veal and | Vas| , Vea > O x a xt x XX vy y = r RG > @ PROJECTILE MOTION © UNIFORM CIRCULAR MOTION 2 + Equation of trajectory y = x tan@, - oe In uniform circular motion a particle moves 2v,5 cos*O, ith a This is equation of parabola. with constant speed. * Time of flight T = _2Vo sin Go y vy = 0 ° Angular displacement AQ = Arc (PP) aang g Radius * Maximum height h,, = (vo sino)” + Angular velocity w = A6 , % n nv 2g At i 2. * Linear speed v = wr * Horizontal range R = Vo sin 28, 2 g * Centripetal acceleration a, = = = mr 2 : . - lho Roe, SAE and a is always directed towards centre of the circle. 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