Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi , Ph. : ,

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1 PK K I N E M A T I C S Syllabs : Frame of reference. Motion in a straight line : Position-time graph, speed and velocity. Uniform and non-niform motion, average speed and instantaneos velocity. Uniformly accelerated motion, velocity-time, position-time graphs, relations for niformly accelerated motion. Scalars and Vectors addition and Sbtraction, Zero Vector,Scalar and Vector prodcts, Unit Vector, Resoltion of a Vector. Relative Velocity, Motion in a plane, Projectile Motion. Einstein Classes, Unit No., 3, Vardhman Ring Road Plaza, Vikas Pri Etn., Oter Ring Road New Delhi 8, Ph. : , 857

2 PK C O N C E P T S CA CB DISPLACEMENT & DISTANCE Displacement is defined as the change in position vector of the particle dring a time interval whereas distance is defined as the length of actal path. Displacement is a vector qantity whereas distance is a scalar qantity. VELOCITY AND SPEED Average Velocity : The change in position vector i.e. displacement divided by time interval dring which this change occrs is known as average velocity. For eample, a particle changes its position from i to f along - ais at time t i and t f respectively. Then average velocity along -ais is given by : v av f i t t f i t r dr Instantaneos Velocity : It is given by v lim, v t t dt d, v dt Average Speed : The average speed of a particle in a time interval is defined as the distance travelled by the particle divided by the time interval. Instantanos Speed : The instantaneos speed eqals the magnitde of the instantaneos velocity. The instantaneos speed is given by Practice Problems : s v lim t t. Which of the following statement is tre? displacement distance Average velocity Average speed ds dt distance and average speed never be zero or negative all the above y dy dt, where s is the distance travel dring time t.. A train travels from one station to another at a speed of v and retrns to the first station at the speed of v. The average speed and average velocity of the train is respectively vv, v v vv, v v, vv v v vv, v v 3. A particle covers one qarter of a circlar path of radis R. It takes time T. The average speed and the magnitde of average velocity are given by respectively. R T, R T R R, T T R T, R T R R, T T [Answers : () d () a (3) a] C ACCELERATION Average Acceleration : Average acceleration is defined as the ratio of change in velocity to the time taken. v vf vi a aav, where vf and v are the velocity of the particle at t i f (final time) and t i t t t (initial time) respectively. f i For straight line motion (i.e. along -ais) a av vf vi v. t t t f i Einstein Classes, Unit No., 3, Vardhman Ring Road Plaza, Vikas Pri Etn., Oter Ring Road New Delhi 8, Ph. : , 857

3 Instantaneos Acceleration : Instantaneos acceleration is defined as a dv dt PK 3 For straight line motion (i.e. along -ais) a dv dt Acceleration can also be epressed as a For niform velocity a. dv d d. dt v C3 Flow chart to find displacement, velocity & acceleration : dv d Practice Problems :. A particle moves along a straight line sch that its displacement at any time t is given by (t 3 3t + )m. The displacement when the acceleration is zero m m 3 m m. The initial velocity of a particle is and the acceleration at time t is at, a being a constant. Then the velocity v at time t is given by v = v = + at v = + at v = + at 3. The displacement of a particle moving in one dimension nder constant acceleration is related to the time t as t = + 3. The displacement of the particle when its velocity is zero is zero 3 nits 3 nits 9 nits 4. The velocity of a particle moving on the -ais is given by v = + where v in m/s and is in m. Its acceleration in m/s when passing throgh the point = m. 5 3 [Answers : () a () d (3) a (4) d] C4 GRAPHICAL REPRESENTATION. The average velocity between two points A and B is the slope of line AB, whereas the instantaneos velocity of the particle at P is the slope of tangent drawn at this point. Einstein Classes, Unit No., 3, Vardhman Ring Road Plaza, Vikas Pri Etn., Oter Ring Road New Delhi 8, Ph. : , 857

4 PK 4. Consider the velocity time graph for a particle moving along the straight line as shown in figre. Let the magnitde of area of the triangle OAB is A and BCD is A then Distance = A + A Magnitde of displacement = A A 3. The average acceleration between two points A and B is the slope of line AB, whereas th e instantaneos acceleration of the particle at P is the slope of tangent drawn at this point. Practice Problems :. The velocity of a car moving along straight road is changing with time as shown in figre Then : The maimm acceleration of the car is between 4s to 5s. The total distance covered by the car is 65 m The total displacement covered by the car is 3 m Dring the jorney there is always non-niform motion. [Answers : () a] Einstein Classes, Unit No., 3, Vardhman Ring Road Plaza, Vikas Pri Etn., Oter Ring Road New Delhi 8, Ph. : , 857

5 C5 PK 5 Some typical graph : In the following graphs time is on the horizontal ais whereas displacement or velocity on the vertical ais C6 MOTION WITH CONSTANT ACCELERATION, ALONG STRAIGHT LINE OR RECTELINEAR MOTION For a niformly accelerated motion along a straight line (sav -ais) the following eqations can be sed. Einstein Classes, Unit No., 3, Vardhman Ring Road Plaza, Vikas Pri Etn., Oter Ring Road New Delhi 8, Ph. : , 857

6 = + (t t ) + ½ a (t t ) v = + a (t t ) PK 6 v = + (t t ) v = + a ( ) The symbols sed above have following meaning; Initial position of the particle on -ais at initial time t. Initial velocity of the particle along -ais. v Velocity of the particle at any position and any time t. a Constant acceleration of the particle along -ais. NOTE : we mst decide at the beginning of a problem where the origin of co-ordinates is and which direction is positive. The choices of frame of reference are sally a matter of convenience. Practice Problems :. A particle starts with velocity along a straight line path with constant acceleration. It ends its jorney with velocity v. The velocity of the particle at the mid point of the jorney is v v v v v v [Answers : () b] C7 Vertical Motion Under Gravity If a body is moving vertically downwards or pwards, it eperiences a downward acceleration de to the gravitational force of the earth. This is called acceleration de to gravity and is denoted by the symbol g. Strictly speaking g is not a constant, bt varies form place to place on the srface of the earth and also with height. However the variation of g is so small that it can be neglected and g can be considered a constant nless very large heights are involved. Therefore, we can se the above eqations of motion for constant acceleration. For solving problems of vertical motion nder gravity, either the pward or the downward direction is taken as positive. If the pward direction is taken as positive, then g becomes negative and vice-versa. The signs of other qantities like initial velocity, initial position will be decided according to the frame of reference. Practice Problems :. A stone is dropped from the top of a 3 m high cliff. At the same instant another stone is projected vertically pwards from the grond with a speed of 3 m/s. The two stones will cross each other after a time t and the height it which they cross each other is h then (g = m/s ) t = s, h = 5 m t = s, h = 5 m t = s, h = 5 m t = s, h = 5 m. A particle, dropped from a height h, travels a distance 9h/5 in the last second. If g = 9.8 m/s, then h is m.5 m 45 m 67.5 m [Answers : () b () b] C8A MOTION IN A PLANE OR D MOTION If a particle is moving in a plane, its motion can be split into two rectilinear motions along two perpendiclar directions. These two motions can be treated independently of each other and then the reslts can be combined according to the rles of vector addition & reqirement of the problem. Now, if the acceleration is constant, then the motions along the two aes are governed by the following two sets of eqations : Einstein Classes, Unit No., 3, Vardhman Ring Road Plaza, Vikas Pri Etn., Oter Ring Road New Delhi 8, Ph. : , 857

7 X-direction Y-direction = + (t t ) + ½ a (t t ) y = y + y (t t ) + ½ a y (t t ) v = + a (t t ) v y = y + a y (t t ) PK 7 C8B v y v y = + (t t ) y = y + (t t ) v = + a ( ) v y = y + a y (y y ) Horizontal projection Sppose a body is projected horizontally from a certain height h with a speed then time of flight = h T and the horizontal range = g R T h g C8C Obliqe Projection Sppose a body is projected with initial velocity at an angle with the horizontal. g (i) The eqation of the trajectory of the projectile is y (tan ) cos which represents a parabola. (ii) Maimm Height H sin g (iii) Timeof Flight T sin g (iv) Horizontal Range R sin g Two important points to be noted concerning horizontal range R : (i) For a given velocity of projection, R is maimm when = 45. (ii) For a given velocity, there are two angles of projection for which the range is the same. If one of these angles is, the other is Practice Problems :.. The and y coordinates of a particle at any time t are given by = 3t + 4t and y = 4t where and y are in m and t in s. Then The initial speed of the particle is 5 m/s. The acceleration of the particle is constant. The path of the particle is parabolic. All are correct. A particle is projected with speed at an angle of with the horizontal. Another particle of different mass is projected with same speed from the same point. Both the particles has same horizontal range. Let the time of flight and maimm height attained by the first particle and second particle are t, h and t, h respectively. Then t /t and h /h are given by respectively tan, tan cot, cot cot, tan tan, cot 3. Let the maimm height attained by the projectile is n times the horizontal range. Then the angle of projection with the horizontal is given by tan n tan n tan 3n tan 4n Einstein Classes, Unit No., 3, Vardhman Ring Road Plaza, Vikas Pri Etn., Oter Ring Road New Delhi 8, Ph. : , 857

8 PK 8 4. Two projectiles are projected from the same point with the same speed bt at different angles of projection. Neglect the air resistance. They land at the same point on the grond. Which of the following angle of projections is possible?, 4 4, 3 6, all are possible 5. If y = a b is the path of a projectile, then which of the following is correct Range = a/b Maimm height = a /4b Angle of projection = tan a all are correct [Answers : () d () a (3) d (4) d (5) d] C9 RELATIVE MOTION If AB is is position of A with respect to B then AB = A B where A and B are the position of A and B with respect to some common frame of reference. In the similar way for relative velocity v AB = v A v B. INITIAL STEP EXERCISE. A motor car is going de (towards) north at a speed of v. It makes a 9 left trn withot changing the speed. The change in the velocity of the car is abot v towards west v towards soth-west v towards north-west zero. Water drops fall at reglar intervals from a roof. At an instant when a drop is abot to leave the roof, the separations between sccessive drops below the roof are in the ratio : : 3 : 4 : 4 : 9 : 6 : 3 : 5 : 7 : 5 : 3 : 3. A point moves in -y plane according to the law = 4 sin 6t and y = 4( cos 6t). The distance traversed by the particle in 4 seconds is ( and y are in meters) 96 m 48 m 4 m 8 m 4. A ball is dropped vertically from a height d above the grond. It hits the grond and bonces p vertically to a height d/. Neglecting sbseqent motion and air resistance, its velocity v varies with the height h above the grond as figre Einstein Classes, Unit No., 3, Vardhman Ring Road Plaza, Vikas Pri Etn., Oter Ring Road New Delhi 8, Ph. : , 857

9 5. An object of mass m is projected with a momentm p at sch an angle that its maimm height (H) is /4th of its horizontal range (R). The ratio of maimm kinetic energy to minimm kinetic energy in its path will be 8 : : 4 : 3 3 : 6. The acceleration vector of a particle is a constant. The trajectory of the particle is a/an parabola ellipse hyperbola circle 7. A hot air balloon is ascending at the rate of m/s and is 4m above the grond when a ball is dropped over the side. The average speed and average velocity of the ball over the whole time of flight are respectively 5.5 m/s, 9.5 m/s, 9.5 m/s.5 m/s, m/s 6.5 m/s,.5 m/s 8. A body is in straight line motion with an acceleration given by a = 3 4v. At t = the velocity of the particle is 4 nit. The velocity when t = ln is 5/ 7/ 3/4 3/4 9. A vector a is trned throgh abot its initial point. The magnitde of change in vector a is a a sin / a cos/ PK 9. Rain is falling with a speed of 4 m/s in a direction making an angle of 3 with vertical towards soth. What shold be the magnitde and direction of velocity of cyclist to hold his mbrella eactly vertical, so that rain does not wet him m/s towards north 4 m/s towards soth m/s towards soth 4 m/s towards north. A stone is projected from the grond with a velocity of 5 m/s at an angle 3. It crosses the wall after 4s. The distance beyond the wall at which the stone strikes the grond is 5 m 53 m 5 m 5/3 m 3. The deceleration eperienced by a moving motor-boat, after its engine is ct off is given by kv 3, where k is a constant. If v is the magnitde of the velocity at ct-off, the magnitde of the velocity at a time t after the ct-off is v v kt v v kt v e kt v e kt 4. The deceleration eperienced by a moving motor-boat, after its engine is ct off is given by kv, where k is a constant. If v is the magnitde of the velocity at ct-off, the magnitde of the velocity at a time t after the ct-off is v v kt v v kt v e kt v e kt 5. The deceleration eperienced by a moving motor-boat, after its engine is ct off is given by kv, where k is a constant. If v is the magnitde of the velocity at ct-off, the maimm distance convered by the boat is. A particle is projected from the top most point of the minar which is at the height of 4 m from its base. The velocity of the particle is m/s at an angle of 6 with the vertical. The particle lands the grond at a distance of from the base of the minar. The vale of is 3 m 3 m 43 m 53 m v k v k v k v 4k Einstein Classes, Unit No., 3, Vardhman Ring Road Plaza, Vikas Pri Etn., Oter Ring Road New Delhi 8, Ph. : , 857

10 6. A stone is dropped from a height h, simltaneosly, another stone is thrown p from the grond which reaches a height 4h. The two stones cross each other after time h g h 8g 8 hg hg 7. A body is projected at time t = from a certain point on a planet s srface with a certain velocity at a certain angle with the planet s srface (assmed horizontal). The horizontal and vertical displacements and y (in metres) respectively vary with time t (in seconds) as = 3 t y = t t PK What is the magnitde and direction of the velocity with which the body is projected? ms at an angle of 3 with the horizontal ms at an angle of 6 with the horizontal ms at an angle of 3 with the horizontal ms at an angle of 6 with the horizontal FINAL STEP EXERCISE. A bird flies for 4 sec with a velocity of (t ) m/s in a straight line, where t = time in seconds. The average speed and average velocity of the bird are,, m/s m/s, m/s, m/s. A river is flowing from west to east at a speed of. A man on the soth bank of the river, capable of swimming at v with respect to river. The width of the river is l. Choose the correct statement. If the man wants to swim across the river in the shortest time, he shold swim de north. If the man wants to swim across the river in the shortest distance, he shold swim de north. If the man wants to swim across the river in the shortest distance, he shold swim sin north of west. v and are correct. 3. A projectile has a maimm range of 5 m. If the projectile is now thrown p an inclined plane of 3 with the same velocity, the distance covered by it along the inclined plane will be abot 5 m 5 m 75 m m 4. Three particles starts from the origin at the same time, one with a velocity along the -ais, the second along the y-ais with a velocity and the third along the = y line. The velocity of the third so that the three may always lie on the same line is 5. Choose the correct statement from the following for a projectile projected from the grond at certain angle with the horizontal. The angle between the velocity vector and acceleration vector at the highest point is /. The minimm speed at the highest point eqals to the initial horizontal speed. The maimm horizontal range for the projectile is at the angle of projection of /4. All are correct. 6. The graph between the displacement and time t for a particle moving in a straight line is shown in the diagram. Dring the intervals OA, AB, BC and CD the acceleration of the particle is Einstein Classes, Unit No., 3, Vardhman Ring Road Plaza, Vikas Pri Etn., Oter Ring Road New Delhi 8, Ph. : , 857

11 OA AB BC CD The displacement () of a particle depends on time (t) as = t t 3. The particle will retrn to its starting point after time / The particle will come to rest after time /3. The initial velocity of the particle was zero bt its initial acceleration was not zero. all are correct 8. A particle starts from the origin of coordinates at time t = and moves in the y plane with a constant acceleration in the y-direction. Its eqation of motion is y =. Its velocity component in the -direction is variable 9. A projectile is projected with speed at an angle with the horizontal. The time after which the velocity vector of the particle become perpendiclar to the initial velocity of projection g sin sin g g cos sin g. A car accelerated from rest at a constant rate for some time after which it decelerates at a constant rate to come to rest. Let the total time is T for the jorney. The maimm velocity attained and total distance covered is respectively T T and and T ( ) ( ) T T and ( ) ( ) T PK. A body is projected vertically pwards with velocity. If t and t be the times at which it is at height h above the point of projection while ascending and descending respectively, then gt t, g(t t ) h h gtt, g(t t ) h gtt, g(t t ) h gtt, g(t t ) ANSWERS (INITIAL STEP EXERCISE). b. c 3. a 4. a 5. b 6. a 7. c 8. d 9. c. c. c. b 3. a 4. b 5. a 6. b 7. a ANSWERS (FINAL STEP EXERCISE). c. d 3. a 4. d 5. d 6. b 7. d 8. d 9. a. a. d T ( ) and T ( ) Einstein Classes, Unit No., 3, Vardhman Ring Road Plaza, Vikas Pri Etn., Oter Ring Road New Delhi 8, Ph. : , 857

12 AIEEE ANALYSIS [3] 3. The coordinates of a moving particle at any time t are given by = t 3 and y = t 3. The speed of the particle at time t is given by t AIEEE ANALYSIS [4/5] 3t 3t 4. A boy playing on the roof at a m high bilding throws a ball with a speed of m/s at an angle of 3 with the horizontal. How far from the throwing point will the ball be at the height of m form the grond? g m / s,sin 3,cos3.6 m 8.66 m 5. m 4.33 m 3 5. A ball is released from the top of a tower of height h metres. It takes T seconds to reach the grond. What is the position of the ball in T/3 seconds? AIEEE ANALYSIS []. From a bilding two balls A and B are thrown sch that A is thrown pwards and B downwards (both vertically with same speed). If v A and v B are their respective velocities on reaction the grond, then v B > v A v A = v B v A > v B their velocities depends on their masses. Two forces are sch that the sm of their magnitdes is 8 N and the magnitde of their resltant is N. If the resltant is perpendiclar to the smaller forces, then the magnitdes of the forces are N, 6 N 3 N, 5 N N, 8 N 6 N, N h/9 metres from the grond 7h/9 metres from the grond 8h/9 metres from the grond 7h/8 from the grond [4] PK 6. A projectile can have the same range R for two angles of projection. If T and T be the time of flights in the two cases, then the prodct of the two time of flights is directly proportional to /R /R R R [4] 7. Which of the following statements is false for a particle moving in a circle with a constant anglar speed? The velocity vector is tangent to the circle The acceleration vector is tangent to the circle The acceleration vector points to the centre of the circle The velocity and acceleration vectors are perpendiclar to each other [4] 8. A ball is thrown from a point with a speed v at an angle of projection. From the same point and at the same instant a person starts rnning with a constant speed v / to catch the ball. Will the person be able to catch the ball? If yes, what shold be the angle of projection? yes, 6 yes, 3 no yes, 45 [4] 9. The relation between time t and distance is t = a + b where a and b are constants. The acceleration is av 3 av abv bv 3 [5]. A car, starting from rest, accelerates at the rate f throgh a distance S, then contines at constant speed for time t and then decelerates at the rate f to come to rest. If the total distance travelled is 5 S, then S ft S = ft S ft 4 S ft 6 [5] Einstein Classes, Unit No., 3, Vardhman Ring Road Plaza, Vikas Pri Etn., Oter Ring Road New Delhi 8, Ph. : , 857

13 . A particle is moving eastwards with a velocity of 5 ms. In seconds the velocity changes to 5 ms northwards. The average acceleration in this time is zero ms ms ms towards north-west towards north-east towards north [5]. A projectile can have the same range R for two angles of projection. If t and t be the times of flight in the two cases, then the prodct of the two time of flights is proportional to R R R R [5] PK 3 3. A parachtist after bailing ot falls 5 m withot friction. When parachte opens, it decelerates at m/s. He reaches the grond with a speed of 3 m/s. At what height, did he bail ot? 93 m m 9 m 8 m AIEEE ANALYSIS [6] 4. A particle located at = at time t =, starts moving along the positive -direction with a velocity v that varies as v = a. The displacement of the particle varies with time as t ½ t 3 t t AIEEE ANALYSIS [7] 5. The velocity of a particle is v = v + gt + ft. If its position is = at t =, then its displacement after nit time (t = ) is v + g/ + f v + g/ + f/3 v + g + 3f v + g + f [5] ANSWERS (AIEEE ANALYSIS). b. b 3. d 4. b 5. c 6. c 7. b 8. a 9. a. a. b. b 3. a 4. c 5. b TEST YOURSELF. The displacement of a particle varies with time a bt according to the relation ( e ). Then b At t = /b, the displacement of the particle is nearly (/3) (a/b) The velocity and acceleration of the particle at t = are a and ab respectively The particle cannot reach a point at a distance from its starting position if > a/b All are correct. A parachtist drops freely from an aeroplane for s before the parachte opens ot. Then he descends with a net retardation of.5 ms. If he bails ot of the plane at a height of 495 m and g = ms, his velocity on reaching the grond will be.5 ms 7.5 ms 5 ms ms 3. A body, starting from rest, moves in a straight line with a constant acceleration a for a time interval t dring which it travels a distance s. It contines to move with the same acceleration for the net time interval t dring which it travels a distance s. The relation between s and s is s = s s = s s = 3s s = 4s Einstein Classes, Unit No., 3, Vardhman Ring Road Plaza, Vikas Pri Etn., Oter Ring Road New Delhi 8, Ph. : , 857

14 4. A body moving in a straight line with constant acceleration of ms covers a distance of 4 m in the 4th second. How mch distance will it cover in the 6th second? 5 m 6 m 7 m 8 m 5. A car, starting from rest, is accelerated at a constant rate ntil it attains a speed v. It is then retarded at a constant rate ntil it comes to rest. The average speed of the car dring its entire jorney is zero v PK 4 v 6. The distance covered by a body moving in a straight line in time t is given by = t + t + 3. The acceleration of the body will vary as 3 v 4 7. Two stones are thrown p simltaneosly with initial speed of and ( > ). They hit the grond after 6s and s respectively. Which graph in figre correctly represents the time variation of = ( ), the relative position of the second stone with respect to the first pto t = s? Assme that the stones do not rebonds after hitting the grond. 8. A person aims a gn at a target located at a horizontal distance of m. If the gn imparts a horizontal speed of 5 ms to the bllet, at what height above the target mst he aim his gn in order to hit it? Take g = ms. cm cm 5 cm cm 9. An aircraft is flying at a niform speed v ms. If the angle sbtended at an observation point on the grond by two positions of the aircraft t seconds apart is, the height of the aircraft above the grond is given by vt tan vt tan vt tan vt tan. A projectile has a range R and time of flight T. If the range is dobled (by increasing the speed of projection, withot changing the angle of projection), the time of flight will become T T T T ANSWERS. d. c 3. c 4. b 5. d 6. c 7. a 8. b 9. d. b Einstein Classes, Unit No., 3, Vardhman Ring Road Plaza, Vikas Pri Etn., Oter Ring Road New Delhi 8, Ph. : , 857