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1 Obliqe Projection. A body is projected from a point with different angles of projections 0 0, 35 0, 45 0, 60 0 with the horizontal bt with same initial speed. Their respective horizontal ranges are R, R, R 3 and R 4. Identify the correct order in which the horizontal ranges are arranged in increasing order ) R, R 4, R, R 3 ) R, R, R 4, R 3 3) R, R, R 4, R 3 4) R 4, R, R, R 3. Two particles are projected from the same point with the same speed at different angles θ and θ to the horizontal. If their respective times of flights are T and T and horizontal ranges are same then a) θ+θ= 90 0 b) T = T tan θ c) T = T tan θ d) T sin = θ T sin θ ) a, b, d are correct ) a, c, d are correct 3) b, c, d are correct 4) a, b, c are correct 3. Two bodies are projected at angles 30 0 and 60 0 to the horizontal from the grond sch that the maimm heights reached by them are eqal. Then a) Their times of flight are eqal b) Their horizontal ranges are eqal c) The ratio of their initial speeds of projection is 3: d) Both take same time to reach the maimm height.

2 Mark the answer as ) If a, b, c and d are correct ) If only a, b and c are correct 3) If only a and c are correct 4) If a, c and d are correct 4. A): A metal ball and a wooden ball of same radis are dropped from the same height in vacm reach the grond same time. R): In vacm all the bodies dropped from same height take same time to reach the grond. ) Both (A) and (R) are tre and (R) is the correct eplanation of (A). ) Both (A) and (R) are tre and (R) is not the correct eplanation of (A). 3) (A) is tre bt (R) is false. 4) (A) is false bt (R) is tre. 5. A): The path followed by one projectile as observed by another projectile is a straight line in niform gravitation field. R): The relative velocity between two projectiles at a given place does not change with time, becase their relative acceleration is zero. ) Both (A) and (R) are tre and (R) is the correct eplanation of (A). ) Both (A) and (R) are tre and (R) is not the correct eplanation of (A). 3) (A) is tre bt (R) is false. 4) (A) is false bt (R) is tre.

3 6. A): If a body is projected obliqely at angle above horizontal with initial speed then its speed at the instant when its velocity makes an angle above the horizontal is R): Horizontal component of velocity of projectile remains constant. ) Both (A) and (R) are tre and (R) is the correct eplanation of (A). ) Both (A) and (R) are tre and (R) is not the correct eplanation of (A). 3) (A) is tre bt (R) is false. 4) (A) is false bt (R) is tre. 7. (A): In case of projectile the angle between velocity and acceleration changes from point to point. (R): Becase it s horizontal component of velocity remains constant while vertical component of velocity changes from point to point de to acceleration de to gravity. () Both (A) and (R) are tre and (R) is the correct eplanation of (A). () Both (A) and (R) are tre and (R) is not the correct eplanation of (A). (3) (A) is tre bt (R) is false. (4) (A) is false bt (R) is tre. 8. Angle between velocity and acceleration vectors in the following cases. List - I List - II a) Vertically projected body e) 90 0 b) For freely falling body h) 80 0 c) For projectile f) Changes from point to point d) In niform circlar motion g) Zero ) a h ; b g ; c f ; d e ) a f ; b g ; c h ; d e 3) a e ; b f ; c h ; d g 4) a g ; b h ; c e ; d f

4 9. A body of mass m is projected with velocity at an angle of 45 o with the horizontal. Match the epressions for its kinetic energy, potential energy, linear momentm and anglar momentm at the top of trajectory. Qantities Epressions (a) Total energy (e) mu 4 g (b) Potential energy (f) m/ (c) Linear momentm (g) m / (d) Anglar momentm () a-h, b-g, c-e, d-f () a-h, b-g, c-f, d-e (3) a-g, b-h, c-e, d-f (4) a-g, b-h, c-f, d-e 0. A body is projected at an angle of 45 o with the horizontal on a flat grond, the angle is the made by the velocity vector and the acceleration de to gravity at any instant match the vales of for varios sitations. Angle (a) 45 o (b) 90 o (c) 35 o (d) 80 o Sitation (h) m 4 (e) At the time of lanching (f) At the same time of striking the grond (g) Vertical projection at the time of lanching (h) At the top of the trajectory () a-f, b-h, c-e, d-g () a-f, b-g, c-e, d-h (3) a-e, b-g, c-f, d-h (4) a-e, b-h, c-f, d-g

5 . Two objects are projected with the same velocity at complimentary angle (θ, 90 - θ ) of projection. List -I List - II (a) The ratio of their ranges (e) tan θ : (b) The prodct of their times of flight (f) g (c) The ratio of their maimm heights (g) : (d) Sm of their maimm heights (h) R g () a-e, b-f, c-g, d-h () a-g, b-e, c-g, d-f (3) a-g, b-h, c-e, d-f (4) a-e, b-f, c-g, d-h. In the presence of heavy atmospheric resistance, the parameters pertaining to projectile s motion is affected as follows. (a) Its maimm height is increased. (b) Its range is redced (c) Its total time of flight is increased (d) Its striking angle is decreased () b, c () a, d (3) a, c (4) b, d 3. A projectile of mass m is fired with velocity v at an angle θ to the horizontal from a point P. Neglecting air resistance, the magnitde of change of momentm between the leaving point P and the arriving point Q at the same level is () mv/ () mv cosθ (3) mvsinθ (4) mv tanθ

6 4. Figre shows for paths for a kicked football. Ignoring the effects of air on the flight, rank the paths according to initial horizontal velocity component, highest first (),, 3, 4 (), 3, 4, (3) 3, 4,, (4) 4, 3,, 5. The path of a projectile in the absence of air drag is shown in the figre by dotted line. If the air resistance is not ignored then which one of the path shown in the figre is appropriate for the projectile? () B () A (3) D (4) C 6. A body is projected with a velocity 60 ms - at 30 0 to horizontal. Its initial velocity vector is ) ^ 0i j ) ^ ^ ^ 30i j 3) 30 3 i + 30 j 4) 30 3 i 7. A body is projected with velocity sch that its horizontal range and maimm vertical heights are same. The maimm heights is g y y A B C D 6 ) ) 4g 3) 7g 4) ^ ^ 8 7 g ^

7 8. A body is projected at an angle 30 to the horizontal with a speed of 30 ms. The angle made by the velocity vector with the horizontal after.5 s is (g=0 ms ) ) Zero ) ) ) Two bodies are thrown from the same point with the same velocity of 50ms. If their angles of projection are complimentary angles and the difference of maimm heights is 30m, their maimm heights (g=0ms - ) ) 50m and 80m ) 47.5m and 77.5m 3) 30m and 60m 4) 5m and 55m 0. A particle is thrown with a velocity at an angle θ from the horizontal. Another particle is thrown with the same velocity at an angle a from the vertical. The ratio of times of flight of the two particles will be ) Tan θ : ) Cot θ: 3) Tan θ : 4) Cot θ :. The horizontal and vertical displacement of a projectile are given by = t and y = 6t - 5t, all the qantities being measred in S.I. system. The maimm height of the projectile is (g=0ms - ) ) 5.6 m ).8 m 3) 64 m 4) 6.4 m 5. The eqation of trajectory of a projectile is y = 0 9 0ms the range of projectile (in meters) is [05 E] ) 36 ) 4 3) 8 4) 9. If we assme g =

8 3. The speed of a projectile at its maimm height is 3 times its initial speed. If the range of the projectile is p times the maimm height attained by it, then p= ) 4/3 ) 3) 4 4) 3/4 4. Two bodies are thrown with the same initial velocity at angles α and (90- α ) to the horizon. What is the ratio of the maimm heights reached by the bodies? ) cot α ) tan α 3) sec α 4) cos α 5. A projectile is thrown at an angle of 30 with a velocity of 0m/s. the change in velocity dring the time interval in which it reaches the highest point is ) 0 m/s ) 5 m/s 3) 5 m/s 4) 0m/s 6. A player kicks a foot ball obliqely at a speed of 0ms so that its range is maimm. Another player at a distance of 4m away in the direction of kick starts rnning at that instant to catch the ball. Before the ball hits the grond to catch it, the speed with which the second player has to rn is (g = 0ms ) ) 4 ms - ) 4 ms - 3) 8 ms - 4) 8 ms - 7. A ball A is projected from the grond sch that its horizontal range is maimm. Another ball B is dropped from a height eqal to the maimm range of A. The ratio of the time of flight of A to the time of fall of B is ) : ) : 3) : 4) :

9 8. A particle is projected with velocity gh and at an angle 60 to the horizontal so that it jst clears two walls of eqal height h which are a distance h from each other. The time interval for which the particle travels between these two walls is ) h ) h h 3) 4) g g g 9. A particle is aimed at a mark which is in the same horizontal plane as that of point of projection. If falls 0 m short of the target when it is projected of an angle of 75 and falls 0m ahead of the target when it is projected with an elevation of 45. The angle of projection for which the particle eactly hits this target is (g=0ms - ) ) 3 Sin 4 ) 4 Sin 5 3) tan h g 4) tan - () 30. When a body is projected from a level grond the ratio of its speed in the vertical and horizontal direction is 4: 3. If the velocity of projection is, the time after which, the ratio of the velocities in the vertical and horizontal directions are reversed is ) 7 0g ) 35 0g 3) 9 g 4) 0 g 3. A body of mass kg is projected from the grond with a velocity 0ms at an angle 30 with the vertical. If t is the time in seconds at which the body is projected and t is the time in seconds at which it reaches the grond, the change in momentm in kgms dring the time (t t ) is ) 40 ) ) ) 60

10 3. A projectile has initially the same horizontal velocity as it wold acqire if it had moved from rest with niform acceleration of 3ms for 0.5 mintes. If the maimm height reached by it is 80m then the angle of projection is [g = 0ms - ] ) tan (3) ) tan 3 3) tan 9 4 4) sin A body of mass m projected vertically pwards with an initial velocity reaches a maimm height h. Another body of mass m is projected along an inclined plane making an angle 30 0 with the horizontal and with speed ''. The maimm distance travelled along the incline is ) h ) h 3) h 34. The height y and the distance along the horizontal plane of a projectile on a 4) 4 h certain planet (with no srronding atmosphere) are given by y = (8 t 5t ) meter and 6t projected is (a) 8 m/sec (b) 6 m/sec (c) 0 m/sec = meter, where t is in second. The velocity with which the projectile is (d) Not obtainable from the data 4

11 35. A body of mass m is thrown pwards at an angle θ with the horizontal with velocity v. While rising p the velocity of the mass after t seconds will be (a) ( v cosθ) + ( v sinθ) (b) (c) (d) ( v cosθ v sinθ) gt v + g t (v sinθ) gt v + g t (v cosθ) gt 36. Neglecting the air resistance, the time of flight of a projectile is determined by (a) U vertical (b) U horizontal (c) U = U vertical + U horizontal / (d) U = U( U vertical + U horizontal) 37. A ball is thrown from a point with a speed v o at an angle of projectionθ. 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, o what shold be the angle of projection? (a) Yes, o o 60 (b) Yes, 30 o (c) No (d) Yes, 45 Key ) ) 3) 4 4) 5) 6) 7) 8) 9) 4 0) ) 3 ) 3) 3 4)4 5) 6) 3 7) 4 8) 9) 0) 3 ) ) 3 3) 3 4) 5) 6) 7) 3 8) 9) 30) 3) 3)3 33) 34)3 35)3 36) 37)

12 6. = cosθ y = sinθ Hints 3 = 60 y = 60 = 30 3 i+ 30 j 7. R= H tanθ = 4 8. sinθ = sin θ H = = 7 g g 8 H = 7g 30 5 tanα = α = H+ H = = 5 0 H H = 30 H = 77.5 H 55 = H = 47.5

13 t 0. = t sina g tan a = cosa g sin θ ( sin θ) 6. H = = = =.8m g g A 0 R = = = 8m s B 5/9 3 = cosθ θ = 30 0 Rtanθ = 4H R = PH 4 3 = P P= Δ = 0 V Δ V = sinθ y Δ Vy = 0 = 5ms 6. For θ = 45 R ma 0 0 = = = 40m g 0 T sinθ 0 = = g 0

14 T = S 7. H = gt 8 8. Δ = cosθ Δ t h= Δ t h= gh Δ t Δ h t = Δ t = gh h g R 0 sin( 75) = R + 0 sin( 45) R 0 = R = 30 R Bt θ = 45 R = = 40 g 30 = sin( θ ) g 30 = 40sin( θ) = ( ) θ = ( ) θ sin 3/4 y 4 = Vy y gt 3 = = 3 V 4.sin 3/ 4 4 y - 4gt = 3 U

15 gt = = 4gt 3 7 t = g + y 3 = 5 7 t = 0g y 5 = 9 3. Δ p= mg T 4 6 = + = y 5 = t = g Δ p = ( 0) 0 = 0 3 Δ p = V = at = V = 90 ms - Bt cosθ = 90 H sin θ = = 80 g sin θ = 600

16 sinθ = cosθ = 90 4 Tanθ = 9 ( ) θ = Tan 4/9 h = g mgh = m h sinθ = l l = h dy d =, v = = 6 dt dt 34. v y = 8 0 t dy = dt At the time of projection i.e. v y = 8 and v = 6 v = v + vy = = 0 m/ s 35. Instantaneos velocity of rising mass after t sec will be v = v + v Where = v cos θ = Horizontal component of velocity v y v t v t v = v sinθ gt = = ( v cos θ ) + ( v sin θ gt) = v + g t Vertical component of velocity v sinθ gt 36. Time of flight = sinθ = = g g g y vertical 37. Person will catch the ball if its velocity will be eqal to horizontal component of t y velocity of the ball. v 0 v0 = cosθ cos θ = θ = 60

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

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