Kinematics

KINEMATICS DESCRIPTION OF MOTION IN ONE DIMENSION Motion in a straight line, uniform and non-uniform motion, their graphical representation. Uniformly accelerated motion, and its applications. DESCRIPTION OF MOTION IN TWO AND THREE DIMENSIONS Scalars and vectors, vector addition, a real number, zero vector and its properties. Resolution of vectors. Scalar and vector products, uniform circular motion and its applications projectile motion. INITIAL STEP EXERCISE   1. A train travels from one station to another at a speed of v1 and returns to the first station at the speed of v2. The average speed and average velocity of the train is respectively (a) 2 v1v 2 ,0 v1 + v 2 0, 2 v1v 2 v1 + v 2 (b) (c) 0, 0 2 v1v 2 2 v1v 2 , v1 + v 2 v1 + v 2 (d) 2. A motor car is going due (towards) north at a speed of v. It makes a 900 left turn without changing the speed. The change in the velocity of the car is about (a) (b) (c) (d) √2v towards west √2v towards south-west √2v towards north-west zero 3. A particle is moving along a circular path with a power, directly proportional to time. The tangential force acting on the particle is directly proportional to tn. Then the value of n is (a) (c) 0 1 (b) (d) 1 2 3 2 4. A particle is projected with velocity u at an angle θ with the horizontal in a gravitational field of intensity E. The field intensity is vertical and acting downward. The time after which the tangential acceleration of the particle becomes zero is (a) 2u sin θ E (b) 3u sin θ 2E 5. 6. 7. 8. 9. u sin θ u sin θ 3E E (d) (c) The x and y coordinates of a particle at any time t are given by x = 3t + 4t2 and y = 4t where x and y are in m and t in s. Then (a) The initial speed of the particle is 5 m/s. (b) The acceleration of the particle is constant. (c) The path of the particle is parabolic. (d) All are correct Water drops fall at regular intervals from a roof. At an instant when a drop is about to leave the roof, the separations between successive drops below the roof are in the ratio (a) 1:2:3:4 (b) 1 : 4 : 9 : 16 (c) 1:3:5:7 (d) 1 : 5 : 13 : 21 A point moves in x-y plane according to the law x = 4 sin 6t and y = 4(1 – cos 6t). The distance traversed by the particle in 4 seconds is (x and y are in meters) (a) 96 m (b) 48 m (c) 24 m (d) 108 m A balloon starts rising from the ground with an acceleration of 1.25 m/s2. After 8 s, a stone is released from the balloon. The stone will (a) cover a distance of 40 m (b) have a displacement of 50 m (c) reach the ground in 4 sec (d) begin to move down after being released A particle of mass m thrown up vertically reaches its highest point in time t1 and returns to the ground in further time t2. There is a constant air friction f on the particle opposite to its direction of motion. Let a = f/m, then t1/t2 equals to (a) g+a g − a (b) g−a g 10. g−a g g + a (d) g−a (c) A ball is dropped vertically from a height d above the ground. It hits the ground and bounces up vertically to a height d/2. Neglecting subsequent motion and air resistance, its velocity v varies with the height h above the ground as figure (a) (b) (c) (d) 11. A balloon starts rising from the ground with an acceleration of 1.25 m/s2. After 8s, a stone is released from the balloon. Take the ground as origin and vertical upward direction as negative. The velocity (v) time (t) graph for the particle during the interval t = 8 s to t = 10 s is given by (a) (b) (c) (d) 12. A particle covers one quarter of a circular path of radius R. It takes time T. The average speed and the magnitude of average velocity are given by respectively. (a) 13. πR 2R , 2T T (b) πR πR , 2T 2T (c) 2R 2R , T T (d) 2R πR , T 2T A particle starts with velocity u along a straight line path with constant acceleration. It ends its journey with velocity v. The velocity of the particle at the mid point of the journey is (a) v+u 2 2 vu v+u (b) v2 + u 2 2 2v 2 u 2 v2 + u 2 (c) 14. (d) An object of mass m is projected with a momentum p at such an angle that its maximum height (H) is 1/4th of its horizontal range (R). The ratio of maximum kinetic energy to minimum kinetic energy in its path will be (a) (c) 8:1 4:3 (b) (d) 2:1 3:2 15. The acceleration of a particle moving in a straight line varies with time as shown in figure. The average acceleration of the particle during t = 0 to t = t0 is given by (a) (b) (c) (d) a0 a0 3 a0 2 a0 4 16. The acceleration vector of a particle is a constant. The trajectory of the particle is a/an (a) (c) parabola hyperbola (b) (d) ellipse circle 17. A hot air balloon is ascending at the rate of 10m/s and is 40m above the ground 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 (a) (b) (c) (d) 5.5 m/s, 0 9.5 m/s, 9.5 m/s 12.5 m/s, 10m/s 16.5 m/s, 12.5 m/s parabola (b) ellipse 18. The modulus of the acceleration vector is constant. The trajectory of the particle is a/an (a) (c) 19. hyperbola (d) circle A body is in straight line motion with an acceleration given by a = 32 – 4v. At t = 0 the velocity of the particle is 4 unit. The velocity when t = ln 2 is (a) (c) 15/2 (b) 17/2 23/4 (d) 31/4 r r A vector a is turned through θ about its initial point. The magnitude of change in vector a is r 2a θ (a) 0 (b) r r 2 a sin θ / 2 2 a cosθ / 2 (d) (c) A particle is projected with speed u at an angle of θ with the horizontal. Another particle of different mass is projected with same speed from the same point. Both the particles have same horizontal range. Let the time of flight and maximum height attained by the first particle and second particle are t1, h1 and t2, h2 respectively. Then t1/t2 and h1/h2 are given by respectively (a) (c) tanθ, tan2θ cotθ, tan2θ (b) (d) cotθ, cot2θ tanθ, cot2θ 20. 21. 22. Let the maximum height attained by the projectile is n times the horizontal range. Then the angle of projection with the horizontal is given by (a) (c) tan–1n tan–13n (b) (d) tan–12n tan–14n where v in m/s and x is in m. Its 23. The velocity of a particle moving on the x-axis is given by v = x2 + x acceleration in m/s2 when passing through the point x = 2m. (a) (c) 0 11 (b) (d) 5 30 24. A ball is projected vertically up with an initial velocity. Which of the following graphs represents the KE of the ball? (a) (b) (c) (d) 25. A particle is projected from the top most point of the lighthouse which is at the height of 40 m from its base. The velocity of the particle is 20 m/s at an angle of 600 with the vertical. The particle lands the ground at a distance of x from the base of the lighthouse. The value of x is (a) (c) 10√3 m 40√3 m (b) (d) 20√3 m 25√3 m 26. Rain is falling with a speed of 4 m/s in a direction making an angle of 300 with vertical towards south. What should be the magnitude and direction of velocity of cyclist to hold his umbrella exactly vertical, so that rain does not wet him? (a) (b) (c) (d) 2 m/s towards north 4 m/s towards south 2 m/s towards south 4 m/s towards north 27. A particle is projected with speed u at an angle θ with the horizontal in vertical plane. The radius of curvature of the path traversed by the particle at the highest point of its trajectory is u 2 cos2 θ g u cos θ 3g 2 2 (a) (b) u 2 cos2 θ 2g u 2 cos2 θ 4g (c) 28. (d) A particle of mass m moves along a circle of radius R with a normal acceleration varying with time as an = kt2, where k is a constant. The tangential force acting on the particle is (a) (c) zero (b) (d) m kR 2m kR 3m kR 29. A stone is projected from the ground with a velocity of 50 m/s at an angle 300. It crosses the wall after 4s. The distance beyond the wall at which the stone strikes the ground is (a) (c) 25 m 50 m (b) (d) 25√3 m 25/√3 m 30. Choose the correct statement (a) (b) (c) (d) In uniform circular motion the net acceleration is always directed towards the centre. In non-uniform circular motion the net acceleration is always directed towards the centre. In uniform circular motion the value of tangential acceleration is not zero. In non-uniform circular motion the value of tangential acceleration is zero. speed and kinetic energy momentum and velocity force and acceleration all the above 31. Which of the following quantity is constant in uniform circular motion? (a) (b) (c) (d) 32. Two projectiles are projected from the same point with the same speed but at different angles of projection. Neglect the air resistance. They land at the same point on the ground. Which of the following angle of projections is possible? (a) (c) π π + θ, − θ 4 4 π θ, − θ 2 (b) (d) π π + θ, − θ 3 6 All are possible Solutions:    1. a 2. b 3. a 4. d 5. d 6. c 7. 13. 19. 25. 31. a b d c a 8. 14. 20. 26. 32. c b c c d 9. 15. 21. 27. c c a a 10. 16. 22. 28. a d d b 11. 17. 23. 29. a c d b 12. 18. 24. 30. a d c a FINAL STEP EXERCISE 1. A bird flies for 4 sec with a velocity of (t – 2) m/s in a straight line, where t = time in seconds. The average speed and average velocity of the bird are (a) (c) 2. 0, 0 1m/s, 0 (b) (d) 0, 1m/s 1m/s, 1m/s Four particles A, B, C and D are thrown from the top of a tower. A is thrown straight up with speed u, B is thrown straight down with the same speed u, C is thrown horizontally with the same speed u and D is released from rest. They hit the ground with speed vA, vB, vC and vD respectively and time of flight are tA, tB, tC and tD respectively. Choose the correct statement from the following (a) (c) vA = vB = vC t D = tC (b) (d) tD = tAtB all are correct 3. Two particle are projected simultaneously in the same vertical plane from the same point, with different speeds u1 and u2, making angles θ1 and θ2 respectively with the horizontal. The path followed by one, as seen by the other (as long as both are in flight) is (a) (b) (c) (d) a vertical straight line if u1cosθ1 = u2cosθ2 a straight line if u1cosθ1 ≠ u2cosθ2 a parabola (a) and (b) are correct 4. A particle is thrown with a speed u at an angle θ with the horizontal. When the particle makes an angle φ with the horizontal, its speed becomes v. Then v equals (a) (b) (c) (d) v = u cos θ v = u cos θ. cos φ v = u cos θ.sec φ v = u sec θ.cos φ 5. A river is flowing from west to east at a speed of u. A man on the south bank of the river, capable of swimming at v with respect to river. The width of the river is l. Choose the correct statement. (a) (b) (c) If the man wants to swim across the river in the shortest time, he should swim due north. If the man wants to swim across the river in the shortest distance, he should swim due north. If the man wants to swim across the river in the shortest distance, he should swim ⎛u⎞ sin −1 ⎜ ⎟ ⎝ v ⎠ north of west. (a) and (c) are correct. (d) 6. A projectile has a maximum range of 500 m. If the projectile is now thrown up an inclined plane of 300 with the same velocity, the distance covered by it along the inclined plane will be about (a) (c) 250 m 750 m (b) (d) 500 m 1000 m 7. Three particles starts from the origin at the same time, one with a velocity u1 along the x-axis, the second along the y-axis with a velocity u2 and the third along the x = y line. The velocity of the third so that the three may always lie on the same line is (a) u1 + u 2 2 (b) u1u 2 (c) 8. u 1u 2 u1 + u 2 (d) 2u1u 2 u1 + u 2 The greatest acceleration or deceleration that a train may have is a. The minimum time in which the train can get from one station to the next at a distance s is s a 1 s 2 a 2s a 2 s a (a) (b) (c) 9. (a) (b) (c) (d) 10. (d) If y = ax – bx2 is the path of a projectile, then which of the following is correct? Range = a/b Maximum height = a2/4b Angle of projection = tan–1a all are correct Choose the correct statement from the following for a projectile projected from the ground at certain angle with the horizontal. (a) (b) (c) (d) The angle between the velocity vector and acceleration vector at the highest point is π/2. The minimum speed at the highest point equals to the initial horizontal speed. The maximum horizontal range for the projectile is at the angle of projection of π/4. All are correct. 11. The graph between the displacement x and time t for a particle moving in a straight line is shown in the diagram. During the intervals OA, AB, BC and CD the acceleration of the particle is OA (a) (b) (c) (d) + – + – AB 0 0 0 0 BC + + – – CD + 0 + 0 12. The displacement (x) of a particle depends on time (t) as x = αt2 – βt3. (a) (b) (c) (d) The particle will return to its starting point after time α/β The particle will come to rest after time 2α/3β. The initial velocity of the particle was zero but its initial acceleration was not zero. all are correct 13. A particle starts from the origin of coordinates at time t = 0 and moves in the xy plane with a constant acceleration α in the y-direction. Its equation of motion is y = βx2. Its velocity component in the x-direction is (a) variable (b) 2α β α 2β (c) 14. α 2β (d) Two particles A and B are initially 40 m apart. A behind B. Particle A starts moving with a uniform velocity of 10 m/s towards B. Particle B starting from the rest has an acceleration of 2 m/s2 in the direction of velocity of A. The minimum distance between the two is (a) (c) 15. 20 m 25 m (b) (d) 15 m 30 m At a certain moment a particle moves towards north at a speed of 7 m/s. whereas its acceleration 2.8 m/s2 acting towards south. Choose the correct statement from the following (a) (b) (c) (d) The velocity of the particle at t = 5s is 7m/s towards south. The distance covered by the particle in third second is 3.5 m. The average speed of the particle is 7m/s during t = 0 to t = 5s. 16. The average velocity of the particle from t = 0 to t = 5s is negative. ˆ A particle P is projected from the origin (0, 0) with a velocity of 20 j at t = 0 and another particle Q is projected with a velocity from the origin at t = 5s. There is a uniform acceleration on both particles along the negative y-axis. The particle P crosses the origin once again at t = 4s where as the particle Q crosses the point (–5, 0) at t = 9s. The velocity of the particle Q at t = 5s is (a) (c) 5ˆ i + 20ˆ j 4 4ˆ i + 10ˆ j 5 (b) (d) 4ˆ i + 20ˆ j 5 none 17. The velocity of a car moving along straight road is changing with time as shown in figure Then : (a) The maximum acceleration of the car is between 40s to 50s. (b) (c) (d) 18. The total distance covered by the car is 650 m The total displacement covered by the car is 320 m During the journey there is always non-uniform motion. A projectile is projected with speed u at an angle θ with the horizontal. The time after which the velocity vector of the particle become perpendicular to the initial velocity of projection (a) (c) u g sin θ (b) (d) u g cos θ u sin θ g 2u sin θ g Solutions: 1. 7. 13. c d d 2. 8. 14. d d b 3. 9. 15. d d a 4. 10. 16. c d a 5. 11. 17. d b a 6. 12. 18. a d a ANALYSIS 1. The coordinates of a moving particle at any time ‘t’ are given by x = αt3 and y = βt3. The speed of the particle at time ‘t’ is given by (a) (c) [Ans. : d] 2. A boy playing on the roof at a 10 m high building throws a ball with a speed of 10 m/s at an angle of 300 with the horizontal. How far from the throwing point will the ball be at the height of 10 m form the ground? t 2 α 2 + β2 3t α 2 + β2 (b) (d) α 2 + β2 3t 2 α 2 + β 2 ⎡ 1 3⎤ 2 0 0 ⎥ ⎢g = 10m / s , sin 30 = , cos30 = 2 2 ⎦ ⎣ (a) (c) 2.60 m 5.20 m (b) (d) 8.66 m 4.33 m [Ans. : b] 1. A ball is released from the top of a tower of height h metres. It takes T seconds to reach the ground. What is the position of the ball in T/3 seconds ? (a) (b) (c) (d) [Ans. : c] h/9 metres from the ground 7h/9 metres from the ground 8h/9 metres from the ground 17h/18 from the ground 2. A projectile can have the same range R for two angles of projection. If T1 and T2 be the time of flights in the two cases, then the product of the two time of flights is directly proportional to (a) (c) 1/R2 R (b) (d) 1/R R2 [Ans. : c] 3. Which of the following statements is false for a particle moving in a circle with a constant angular speed? (a) (b) (c) (d) [Ans. : b] 4. A ball is thrown from a point with a speed v0 at an angle of projection θ. From the same point and at the same instant a person starts running with a constant speed v0/2 to catch the ball. Will the person be able to catch the ball? If yes, what should be the angle of projection? (a) (c) [Ans. : a] 5. The relation between time t and distance x is t = ax2 + bx where a and b are constants. The acceleration is (a) (c) [Ans. : a] 6. A car, starting from rest, accelerates at the rate f through a distance S, then continues at constant f speed for time t and then decelerates at the rate 2 to come to rest. If the total distance travelled is 15 S, then –2av3 –2abv2 (b) (d) –2av2 2bv3 yes, 600 no (b) (d) yes, 300 yes, 450 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 perpendicular to each other S= (a) (c) [Ans. : a] 7. 1 2 ft 2 S= (b) 1 2 ft 4 1 2 ft 6 S= S = ft (d) A particle is moving eastwards with a velocity of 5 ms–1. In 10 seconds the velocity changes to 5 ms–1 northwards. The average acceleration in this time is (a) (c) [Ans. : b] zero (b) (d) 1 ms − 2 2 towards north-west 1 ms − 2 2 towards north 1 ms − 2 2 towards north-east 8. A projectile can have the same range ‘R’ for two angles of projection. If ‘t1’ and ‘t2’ be the times of flight in the two cases, then the product of the two time of flights is proportional to (a) (c) [Ans. : b] 9. 1 R R 2 (b) (d) R 1 R2 A parachutist after bailing out falls 50 m without friction. When parachute opens, it decelerates at 2 m/s2. He reaches the ground with a speed of 3 m/s. At what height, did he bail out? (a) (c) 293 m 91 m (b) (d) 111 m 182 m [Ans. : a]