Motion in a Plane PYQs PDF

# Motion in a Plane ## Important Formulae and Shortcut Methods 1. **(i) Average speed = path length / time** 2. **(ii) Average velocity = distance / time** 3. **(i) Instantaneous velocity = dr/dt, v = dr/dt** 4. **(ii) Instantaneous acceleration = dv/dt, a = dv/dt** 5. **For uniform accelerat...

# Motion in a Plane ## Important Formulae and Shortcut Methods 1. **(i) Average speed = path length / time** 2. **(ii) Average velocity = distance / time** 3. **(i) Instantaneous velocity = dr/dt, v = dr/dt** 4. **(ii) Instantaneous acceleration = dv/dt, a = dv/dt** 5. **For uniform acceleration :** * **(i) Vav = 1/2(u + v)** * **(ii) v = u + at** * **(iii) s = vav t = ut + 1/2 at² = vt - 1/2at²** * **(iv) v² = u² + 2as** 6. **For vertical one-dimensional motion under gravity :** * **(i) v = u - gt** * **(ii) s = ut - 1/2 gt²** * **(iii) v² = u² - 2gs** 7. **(i) Area under velocity-time graph gives the displacement of the object.** 8. **(ii) Slope of velocity-time graph gives the acceleration of the object.** 9. **Relative velocity of object A with respect to object B : VBA = VA - VB** 10. **Projectile motion :** * **(i) vx = ux = u cos θ** * **(ii) vy = uy - gt = u sin θ - gt** * **(iii) velocity at any instant v = √vx² + vy² = √(u cosθ)² + (u sin θ - gt)²** * **(iv) x = u cosθ . t** * **(v) y = u sin θ - 1/2 gt² = x tan θ - (g / 2u² cos² θ) x²** * **(vi) Distance between the point of projection and the position at any instant t is given by: r = √x² + y²** * **(vii) Time of flight of a projectile T = 2 u sin θ / g** * **(viii) Maximum height of a projectile H = u² sin² θ / 2g** * **(ix) Horizontal range R = u² sin 2θ / g** * **(x) Horizontal range is maximum when 2θ = 90° or θ = 45° Rmax = u²/g** * **(xi) Horizontal range is same for angles of projection θ and (90° – θ)** ## (2) MHT-CET Exam Questions 1. **Uniform circular motion :** * **(i) T = 1/f = 2π/ω** * **(ii) ω = 2π/T = 2πf** * **(iii) v = rω** * **(iv) a = v²/r = ω²r = vω** * **(v) Centripetal force F = ma = mv²/r = mω²r = mvω** * **(vi) For conical pendulum** * **(a) T = 2π √L cos θ / g** * **(b) tan θ = v²/rg** * **(vii) Centripetal force is directed towards the centre and makes the particle move in a circular path.** * **(viii) Centrifugal force is directed away from the centre and is observed in a non-inertial frame.** ## Multiple Choice Questions **MHT-CET 2005** 1. **Angular velocity of hour hand of a watch is** * **(A) π/43200 rad s-¹** * **(B) π/30 rad s-¹** * **(C) π/21600 rad s-¹** * **(D) 1800 rad s-¹** **MHT-CET 2006** 2. **An electric fan has blades of length of 30 cm as measured from the axis of rotation. If the fan is rotating at 1200 rpm, the acceleration of a point on the tip of the blade is about.** * **(A) 1600ms²** * **(B) 4750ms²** * **(C) 2370 ms-²** * **(D) 5055 ms²** **MHT-CET 2011** 3. **A ball of mass 0.25 kg attached to the end of a string of length 1.96 m is moving in a horizontal circle. The string will break if the tension is more than 25 N. What is the maximum speed with which ball can be moved?** * **(A) 14 ms¹** * **(B) 3 ms¹** * **(C) 3.92 ms¹** * **(D) 5 ms¯¹** **MHT-CET 2014** 4. **The difference between angular speed of minute hand and second hand of a clock is** * **(A) 59π/900 rad/s** * **(B) 59π/1800 rad/s** * **(C) 59π/2400 rad/s** * **(D) 59π/3600 rad/s** **MHT-CET 2016** 5. **Angular speed of hour hand of a clock in degree per second is** * **(A) 1/30** * **(B) 1/60** * **(C) 1/120** * **(D) 1/720** **MHT-CET 2019** 6. **A particle is performing U.C.M. along the circumference of a circle of diameter 50 cm with frequency 2 Hz. The acceleration of the particle in m/s² is** * **(A) 2π²** * **(B) 4π²** * **(C) 8π²** * **(D) π²** 7. **In U.C.M. when time interval 𝛿t → 0, the angle between change velocity (𝛿V) and linear velocity (V) will be** * **(A) 0°** * **(B) 45°** * **(C) 90°** * **(D) 180°** 8. **If the radius of the circular path and frequency of revolution of a particle of mass 'm' are doubled then the change in its kinetic energy will be (E₁ and Ef are the initial and final kinetic energies of the particle respectively)** * **(A) 8E;** * **(B) 15E;** * **(C) 12E,** * **(D) 16 E;** 9. **A body of mass ‘m' is performing a U.C.M. in a circle of radius 'r' with speed ‘v'. The work done by the centripetal force in moving it through 1/3rd of the circular path is** * **(A) mv²π/2** * **(B) 2mv²/3** * **(C) zero** * **(D) 2mv²π/3** 10. **The ratio of the angular speed of the hour hand of a clock to that of its minute hand is** * **(Α) 3600 : 1** * **(B) 1:24** * **(C) 1: 12** * **(D) 12:1** 11. **A wheel completes 2000 revolutions to cover the distance of 9.42 km. The diameter of this wheel is π = 3.14** * **(A) 1 cm** * **(B) 1 m** * **(C) 1.5 cm** * **(D) 1.5 m** 12. **A particle is performing a uniform circular motion along the circumference of a circle of radius 'R' and 'T' is the periodic time. In the time 'T/4' its displacement and distance covered are respectively** * **(A) √2R, πρ/4** * **(B) πρ/2,√2R** * **(C) √2R, πρ** * **(D) √2R, R** 13. **Two stones of masses m and 3m are whirled in horizontal circles, the heavier one in radius 3r/2 and lighter one in radius ‘r’. The tangential speed of lighter stone is 'n' times that of the value of heavier stone, when they experience same centripetal force. The value of n is** * **(A)4** * **(B) 1** * **(C) 2** * **(D)3** **MHT-CET 2020** 14. **A vehicle moving with 15 km/hr comes to rest by covering 5m distance by applying brakes. If the same vehicle moves at 45 km/hr, then by applying brakes, it will come to rest by covering a distance** * **(A) 60 m** * **(B) 15 m** * **(C) 45 m** * **(D) 30 m** 15. **A moving body is covering distances which are proportional to square of the time. Then the acceleration of the body is** * **(A) constant but not zero** * **(B) increasing** * **(C) zero** * **(D) decreasing** 16. **A body is just revolved in a vertical circle of radius ‘R’. When the body is at highest point, the string breaks. The horizontal distance covered by the body after the string breaks is** * **(A) R** * **(B) 4R** * **(C) 2R** * **(D) 3R** **MHT-CET 2019** 17. **A particle is moving along the circular path of radius 'r' with velocity 'v'. The magnitude of average acceleration after half revolution is** * **(A) 3v²/πr** * **(B) 3v²/2πr** * **(C) 2v²/πr** * **(D) v²/πr** 18. **Two particles are performing uniform circular motion about a centre of two concentric circles of radii 'r', and 'r2', respectively. The two particles and the centre of circles lie on a straight line during the motion, then the ratio of their angular velocities will be** * **(A) 3:1** * **(B) 2:1** * **(C) 0.5: 1** * **(D) 1:1** 19. **A particle is moving in uniform circular motion with speed 'V' and radius 'R'. The angular acceleration of the particle is** * **(A) V²/R along tangent to the circle.** * **(B) zero** * **(C) V²/R along the radius towards the centre of the circle** * **(D) V²/R perpendicular to the plane of the circle.** 20. **A stone of mass 3 kg attached at one end of a 2m long string is whirled in horizontal circle. The string makes an angle of 45° with the vertical then the centripetal force acting on the string is (g = 10 m/s², tan 45° = 1)** * **(A) 20 N** * **(B) 30 N** * **(C) 10 N** * **(D) 40 N** 21. **A body of mass ‘m' is moving along a circle of radius 'r' with linear speed 'v'. Now, to change the linear speed to v/2 and to move it along the circle of radius '4r', required change in the centripetal force of the body is** * **(A) decrease by 15/16** * **(B) decrease by 5/16** * **(C) increase by 9/16** * **(D) increase by 11/16** 22. **A string of length 'l' is fixed at one end and carries a mass 'm' at the other end. The mass is revolving along a horizontal circle of radius 'r' making ‘0' as the semi-vertical angle of cone and revolutions per second around the vertical axis through fixed end. The tension in the string is** * **(A) 2 mlω²** * **(B) 8 mlω²** * **(C) 4 mlω²** * **(D) 16 mlω²** 23. **At any instant, the magnitude of the centripetal force on a particle of mass 'm' performing circular motion is given by (ω = angular velocity and v = linear velocity of the particle)** * **(A) mω²/V** * **(B) mv²/r** * **(C) mω²/r** * **(D) mωV** 24. **Mass of 0.5 kg is attached to a string moving in horizontal circle with angular velocity 10 cycle/min. Keeping the radius constant, tension in the string in made 4 times by increasing angular velocity 'ω'. The value 'ω' of that mass will be** * **(A) 1/4 cycle/s** * **(B) 1/2 cycle/s** * **(C) 1 cycle/s** * **(D) 2 cycle/s** 25. **A particle is performing uniform circular motion. If ‘θ', 'ω', 'α' and 'a' are its angular displacement, angular velocity, angular acceleration and centripetal acceleration respectively, then which of the following is ‘WRONG’?** * **(Α) ν ∝ ω** * **(B) ν ∝ α** * **(C) δθ ∝ α** * **(D) δα ∝ α** 26. **A ball of mass ‘m' is attached to the free end of an inextensible string of length 'l'. Let 'T' be the tension in the string. The ball is moving in horizontal circular path about the vertical axis. The angular velocity of the ball at any particular instant will be** * **(A) √Tm/l** * **(B) √T/m** * **(C) √T/ml** * **(D) √ml/T** 27. **Two cars of masses ‘m₁, and 'm₂' are moving in the circles of radii 'r₁' and 'r₂' respectively. Their angular speed ‘ω₁' and 'ω₂' are such that they both complete one revolution in the same time 't'. The ratio of linear speed of ‘m₁' to the linear speed of ‘m₂' is** * **(A) r₁ : r₂** * **(B) T²: T** * **(C) r₁: r₂** * **(D)m₁: m₂** 28. **A string of length 'l' fixed at one end carries a mass 'm' at the other end. The string makes 3/π revolutions/second around the vertical axis through the fixed end as shown in figure. The tension ‘T' in the string is** * **(A) 36 mlω²** * **(B) 3 mlω²** * **(C) 18 mlω²** * **(D) 9 mlω²** 29. **A particle performing U.C.M. of radius 'r' makes 'x' revolutions in time ‘t'. Its tangential velocity is** * **(A) πx/t** * **(B) π²x²/t** * **(C) π²/t x** * **(D) πr/t²** 30. **A mass 'm' is tied to one end of a spring and whirled in a horizontal circle with constant angular velocity. The elongation in the spring is 1 cm. If the angular speed is doubled, the elongation in the spring is 6 cm. The original length of the spring is** * **(A) 3 cm** * **(B) 12 cm** * **(C) 6 cm** * **(D) 9 cm** 31. **The angular speed of the minute hand of a clock in degrees per second is** * **(A) 0.1** * **(B) 10** * **(C) 1** * **(D) 0.1** **MHT-CET 2021** 32. **A particle starting from rest moves along the circumference of a circle of radius r = √2 m with an angular acceleration α = π/2 rad/s². The magnitude of its average velocity in the time it completes a quarter rotation, is** * **(A) π/2√2 m/s** * **(B) π/√2 m/s** * **(C) π/2√2 m/s** * **(D) π/2 m/s** 33. **A body starts falling from height ‘h' and travels a distance h/2 during last second of its motion then time of flight in second is** * **(A) (2+√3)** * **(B) (√2-1)** * **(C) (2+√2)** * **(D) (√2 + √3)** **MHT-CET Exam Questions** 34. **The angular speed of the minute hand of a clock in degrees per second is** * **(A) 0.1** * **(B) 0.4** * **(C) 0.3** * **(D) 0.2** 35. **A particle is moving in a circle of radius 'R' with constant speed 'V'. The magnitude of average acceleration after half revolution is** * **(A) V²/πR** * **(B) 2V²/πR** * **(C) 2V²/πR** * **(D) V²/2πR** 36. **A car travelling at a speed ‘U` m/s, stops within a distance 'S', when the brakes are applied. If the car is travelling at ‘2U' m/s then the stopping distance** * **(A) more than 'S'** * **(B) less than 'S’** * **(C) equal to 'S'** * **(D) zero** 37. **A body of mass ‘m` is tied to one end of a spring and whirled round in a horizontal circle with a constant angular velocity. The elongation in the spring is 1cm. If the angular velocity is doubled, the elongation in the spring is 5 cm. The original length of the spring is** * **(A) 16 cm** * **(B) 14 cm** * **(C) 13 cm** * **(D) 15 cm** 38. **A driver applies the brakes on seeing the red traffic signal 400 m ahead. At the time of applying the brakes, the vehicle was moving with 15 m/s and retarding at 0.3 m/s². The distance covered by the vehicle from the traffic light 1 minute after the application of brakes is** * **(A)25 m** * **(B) 360 m** * **(C) 40 m** * **(D)375 m** 39. **A student is throwing balls vertically upwards such that he throws the 2nd ball when the 1st ball reaches maximum height. If he throws balls at an interval of 3 second, the maximum height of the balls is (g = 10 m/s²)** * **(A) 45 m** * **(B) 35 m** * **(C) 25 m** * **(D) 30 m** 40. **Two bodies 'A' and 'B' start from the same point at the same instant and move along a straight line. 'A' moves with uniform acceleration (a) and ‘B' moves with uniform velocity (V). They meet after time 't'. The value of 't' is** * **(A) 2V/a** * **(B) V/a** * **(C) 2V/a** * **(D) V/2a** 41. **Two bodies A and B are projected with same velocity. If bodies A and B are projected at an angle of 30° and 60° with the horizontal respectively, the ratio of maximum height reached by the body A to that of body B is** * **(A) 1:2** * **(B) 2:1** * **(C) 3:1** * **(D) 1:3** 42. **The engine of an aeroplane during take-off exerts a force of 150 × 10³ N. Mass of aeroplane is 25 × 10³ kg. If the take-off speed is 60 m/s, the length of the runway required is** * **(A) 300 m** * **(B) 100 m** * **(C) 200 m** * **(D) 400 m** 43. **A particle moves in a circular orbit of radius 'r' under a central attractive force, F = -k/r, where k is a constant. The periodic time of its motion is proportional to** * **(A) r¹/2** * **(B) r³/2** * **(C) r** * **(D) r²/2** 44. **A bomb is dropped by an aeroplane flying horizontally with a velocity 200 km/hr and at a height of 980 m. At the time of dropping a bomb, the distance of the aeroplane from the target on the ground to hit directly is (g = 9.8 m/s²)** * **(A) √2×10⁴/9 m** * **(B) 10⁴/9 m** * **(C) 10⁴/9√2 m** * **(D) 10⁴/18 m** 45. **A body at rest falls through a height 'h' with velocity 'V'. If it has to fall down further for its velocity to become three times, the distance travelled in that interval is** * **(A) 8 h** * **(B) 6 h** * **(C) 4 h** * **(D) 12 h** 46. **If ω₁ is angular velocity of hour hand of clock and ω₂ is angular velocity of the earth, then the ratio ω₁:ω₂ is** * **(A) 1:2** * **(B) 2:3** * **(C) 3:2** * **(D) 2:1** 47. **The angular displacement of body performing circular motion is given by θ = 5 sin πt/6. The angular velocity of the body at t = 3 second will be (sin π/2 =1, cos π/2 = 0)** * **(A) 5 rad/s** * **(B) 1 rad/s** * **(C) 2.5 rad/s** * **(D) zero rad/s** 48. **A body performing uniform circular motion of radius 'R' has frequency 'n'. It centripetal acceleration is** * **(A) 8 π²nR²** * **(Β) 4 π²n²R** * **(C) 4 π² n²R²** * **(D) 8 π² n² R** 49. **A projectile is thrown with an initial velocity (ai+bj) m/s, where iandj are unit vectors along horizontal and vertical directions respectively. If the range of the projectile is twice the maximum height reached by it, then** * **(A) b = 2a** * **(B) b = 4a** * **(C) b = a/2** * **(D) b = a** 50. **A particle is moving along the circular path with constant speed and centripetal acceleration 'a'. If the speed is doubled, the ratio of its acceleration after and before the change is** * **(A) 3:1** * **(B) 1:4** * **(C) 2:1** * **(D) 4:1** 51. **A body of mass 'm' is moving with speed 'V' along a circular path of radius 'r'. Now the speed is reduced to V/2 and radius is increased to ‘3r'. For this change, initial centripetal force needs to be** * **(A) increased by 7/12 times.** * **(B) increased by 10/12 times.** * **(C) decreased by 11/12 times.** * **(D) decreased by 1/12 times.** 52. **A stone is projected vertically upwards with velocity ‘V. Another stone of same mass is projected at an angle fo 60° with the vertical with the same speed (V). The ratio of their potential energies at the highest points of their journey, is [sin30° = cos60° = 0.5, cos30° = sin60° = √3/2]** * **(A) 1:1** * **(B) 4:1** * **(C) 3:2** * **(D) 2:1** **MHT-CET 2022** 53. **A body covers half of its distance with speed 'u' and the other half with a speed 'v' the average speed of the body is** * **(A) 2uv/(u+v)** * **(B) (u+v)/2uv** * **(C) (u+v)/2** * **(D) (u-v)/2** 54. **A projectile thrown from the ground has initial speed 'u' and its direction makes an angle 'θ' with the horizontal. If at maximum height from ground, the speed of projectile is half its initial speed of projection, then the maximum height reached by the projectile is [g = acceleration due to gravity, sin 30° = cos 60° = 0·5, cos 30° = sin 60°= √3/2]** * **(A) 2u²/g** * **(B) u⁴/4g** * **(C) 3u²/8g** * **(D) u²/g** 55. **A particle is performing a uniform circular motion along a circle of radius 'R'. In half the period of revolution, its displacement and distance covered are respectively** * **(A) 2R, 2πR** * **(B) 2R, πR** * **(C) √2R, 2πR** * **(D) R, πR** 56. **The relative angular speed of hour hand and minute hand of a clock is (in rad/s)** * **(A) 11π/21600** * **(B) 9π/1860** * **(C) 4π/243** * **(D) 7π/1480** 57. **The angular velocity of the minute hand of a clock in degrees per second is** * **(A) 0.6** * **(B) 0.1** * **(C) 0.24** * **(D) 0.12** 58. **An aeroplane is flying in a horizontal direction with a velocity of 540 km/hr at a height of 1960 m. When it is vertically above the point A on the ground, a body is dropped from it. The body strikes the ground at point B. The distance AB is equal to (g = 9.8 m/s²)** * **(A) 3600 m** * **(B) 3000 m** * **(C) 4000 m** * **(D) 2000 m** 59. **When the bob of mass 'm' moves in a horizontal circle of radius 'r' with uniform speed ‘V’, having length of string 'L' describes a cone of semi vertical angle 'θ'. The centripetal force acting on the bob is given by [g = acceleration due to gravity.]** * **(A) mgr/(L²-r²)** * **(B) mgL/(L²-r²)** * **(C) mg√(L²-r²)/L²** * **(D) mg√(L²-r²)/r²** 60. **A particle performing U.C.M. of radius 'r' m' makes 'x' revolutions in time ‘t'. Its tangential velocity is** * **(A) πx/t** * **(B) πx/t** * **(C) 2πx/t** * **(D) π²x²/t** 61. **A body moving in a circular path with a constant speed has constant ** * **(A) kinetic energy.** * **(B) momentum.** * **(C) velocity.** * **(D) acceleration.** 62. **A shell fired at an angle of 30° to the horizontal with velocity 196 m/s. The time of flight is [sin 30° = 1/2 = cos 60°]** * **(A) 10 s** * **(B) 20 s** * **(C) 6.5 s** * **(D) 16.5 s** 63. **Two bodies of masses ‘m' and '3m' are rotating in horizontal circles of radius 'r' and r/3, respectively. The tangential speed of body of mass 'm' is n times that of the value of heavier body, while the centripetal force is same for both. The value of n is** * **(A) 1** * **(B) 9** * **(C) 6** * **(D) 3** 64. **A ball is dropped from the tower of height 'h'. The total distance covered by it in last second of its motion is equal to the distance covered by it in first 3 seconds. The value of ‘h' is (g = 10 ms²)** * **(A) 200 m** * **(B) 125 m** * **(C) 100 m** * **(D) 80 m** 65. **Two balls A and B are projected at an angle of 45° and 60° respectively so that the maximum heights reached are same for both. The ratio of initial velocity of projection of the ball A to that for ball B is [sin 45° = cos 45° = 1/√2, sin 60° = cos 30°=√3/2]** * **(A) 2:√3** * **(B) √3:√2** * **(C) √2:√3** * **(D) √3:2** 66. **A body is just revolved in a vertical circle of radius 'R'. When the body is at highest point, the string breaks. The horizontal distance covered by the body after the string breaks is** * **(A) R** * **(B) 3R** * **(C) 2R** * **(D) 4R** 67. **A vehicle moving with 15 km/hr comes to rest by covering 5 m distance after applying brakes. If the same vehicle moves at 45 km/hr then by applying brakes it will come to rest by covering a distance** * **(A) 30 m** * **(B) 15 m** * **(C) 60 m** * **(D) 45 m** 68. **Two bodies A and B move in same straight line starting from the same position. Body A moves with constant velocity 'u' and B moves with constant acceleration ‘a'. When their velocities become equal, the distance between them is** * **(A) 2au²/u²** * **(B) u²/2a** * **(C) 2u²/3a** * **(D) 2u²a** 69. **At any instant, for a body performing uniform circular motion, velocity vector and acceleration vector are** * **(A) in opposite direction** * **(B) along the same direction** * **(C) normal to each other** * **(D) make an angle of 45° with each other** 70. **Two bodies are thrown up at angles of 45° and 60° with the horizontal respectively. If same vertical height is attained by both the bodies, then the ratio of velocities with which they are thrown is [sin 45° = cos 45° = 1/√2, sin 60° = cos 30° = √3/2, sin 30° = cos 60° = 1/2]** * **(A) √2/√3** * **(B) √3/√2** * **(C) √2/1** * **(D) √2/√3** 71. **A stone of mass 'm’ is tied to a string of length ‘L’ and moved in a vertical circle at the rate of ‘