# 3-Projectile Motion

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3.1 (C) Using equation of trajectory : gx2 –h = x tan (0°) – 2(2gh)(cos2 0)  x = 2h Ans. Method II time of flight T = horizontal distance covered during time of flight is x = ux t = × = 2h 3.2 (D) Ranges for complementary angles are same  Required angle =  5 13 – = Ans. 2 36 36 3.3 (D) Use  =  = 45° in the formula for Range down the incline plane. 3.4 (D) 2uy Time of flight T = g 2  20 sin 37 3 12 T = 10 = 4 × 5 = 5 sec. 12 Range R = ux × T = 5 × (20 cos 37° + 10) 12 R = 5 4 × (20 × 5 12 + 10) = 26 × 5 = 62.4 m 3.5 (C) Use the given data in the formulae for projection up the inclined plane. Let the inclination of the inclined plane =  u cos  = 10 (1) 2usin Time of flight gcos = 2 (2) maximum height u sin  = 5  u = 5 5 u2 sin2  gcos = 5 (3) u cos  = 10 (1) 2usin gcos u2 sin2  gcos u sin  = 5  u = 5 = 2 (2) = 5 (3) 3.6 (D) Path will not be straight line but parabolic hence neither stone will hit any person. Condition of collision will depend upon direction as well as magnitude of velocities of projection which are not given. 3.7 (B) It can be observed from figure that P and Q shall collide if the initial component of velocity of P on inclined plane i.e along incline. u|| = 0 that is particle is projected perpendicular to incline.  Time of flight T = 2u = 2u gcos  u = gT cos  2 = 10 m/s. 3.8 (C) tan = 9  1 4  0 = 2 Where  is the angle of projection 1 Displacement in y-direction sy = uy t + 2 ay t now, – 1 = usin (1) – 1 g (1)2 2 usin = 4 and from triangle sin  =  u = 2 m/s Displacement in x-direction s = u t + 1 a t2 x x 2 x now, x = u cos (1) = (2 5 ) × = 2m 3.9 (B) Two second before maximum height vy = g × 2 = 20 m/s 20 tan53° = v x vx = 15 m/s velocity at maximum height v = vx = 15 m/s 3.10 (C) For B to C H = 1 g (2t)2 = 2gt2 (1) 2 1 h = 2 g t2 (2) 1 h = H–h  h = H – 2 By (1) & (2) gt2 (3) h = H – H = 3H 4 4 3.11 (A) velocity component ux = 400/3 ˆi , uy = 100 ˆj Applying equation is y direction 1 – 1500 = – 100 t – t2 × 10 t2 2  2 + 10 t – 150 = 0  20  40 t = 2 So t = 10 sec i.e. horizontal distance u × t = 500  4  10 = 4000 m. x 3 5 3 3.12 (B) For minimum number of jumps, range must be maximum. u2 ( 10 )2 maximum range = g = 10 = 1 meter. Total distance to be covered = 10 meter So minimum number of jumps = 10 3.13 (D) y = bx2 dy dx d2y  dx 2 d2 x dt = 2bx. dt  dt = 2b   dt + 2bx 2   dt a = 2bv2 + 0  v = 3.14 (D) Applying equation of motion perpendicular to the incline for y = 0. 1 y = uy t + 2 a t 2 0 = V sin ( )t + 1 2  t = 0 & (– g cos) t2 2V sin(  ) gcos At the moment of striking the plane, as velocity is perpendicular to the inclined plane hence component of velocity along incline must be zero. Vx = ux + ax t 0 = v cos (

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### 3-Projectile Motion

• 1. 3.1 (C) Using equation of trajectory : –h = x tan (0°) – ) 0 )(cos gh 2 ( 2 gx 2 2   x = 2h Ans. Method II time of flight T = g h 2 horizontal distance covered during time of flight is x = ux t = g h 2 × hg 2 = 2h 3.2 (D) Ranges for complementary angles are same  Required angle = 2  – 36 5 = 36 13 Ans. 3.3 (D) Use  =  = 45° in the formula for Range down the incline plane. 3.4 (D) Time of flight T = g u 2 y T = 10 37 sin 20 2   = 4 × 5 3 = 5 12 sec. Range R = ux × T = 5 12 × (20 cos 37° + 10) R = 5 12 × (20 × 5 4 + 10) = 26 × 5 12 = 62.4 m 3.5 (C) Use the given data in the formulae for projection up the inclined plane. Let the inclination of the inclined plane =  u cos  = 10 ......... (1) Time of flight   cos g sin u 2 = 2 .......... (2) maximum height   cos g sin u 2 2 = 5 ............. (3) u sin  = 5  u = 5 5 u cos  = 10 ......... (1) RESONANCE RRBP910PAGE-1 3. PROJECTILE MOTION & VECTOR
• 2.   cos g sin u 2 = 2 .......... (2)   cos g sin u 2 2 = 5 ............. (3) u sin  = 5  u = 5 5 3.6 (D) Path will not be straight line but parabolic hence neither stone will hit any person. Condition of collision will depend upon direction as well as magnitude of velocities of projection which are not given. 3.7 (B) It can be observed from figure that P and Q shall collide if the initial component of velocity of P on inclined plane i.e along incline. u|| = 0 that is particle is projected perpendicular to incline.  Time of flight T =   cos g u 2 =  cos g u 2  u = 2 cos gT  = 10 m/s. 3.8 (C) tan = 0 4 1 9   = 2 Where  is the angle of projection Displacement in y-direction sy = uy t + 2 1 ay t2 now, – 1 = usin (1) – 2 1 g (1)2 usin = 4 and from triangle sin  = 5 2  u = 5 2 m/s Displacement in x-direction sx = ux t + 2 1 ax t2 now, x = u cos (1) = ) 5 2 ( × 5 1 = 2m 3.9 (B) Two second before maximum height vy = g × 2 = 20 m/s tan53° = x v 20 vx = 15 m/s velocity at maximum height v = vx = 15 m/s
• 3. 3.10 (C) For B to C H = 2 1 g (2t)2 = 2gt2 ..............(1) h = 2 1 g t2 ............. (2) h = H–h  h = H – 2 1 gt2 .............(3) By (1) & (2) h = H – 4 H = 4 H 3 3.11 (A) velocity component ux = 400/3 î , uy = 100 ĵ Applying equation is y direction – 1500 = – 100 t – 2 1 × 10 t2  2 t2 + 10 t – 150 = 0 t = 2 40 20   So t = 10 sec i.e. horizontal distance ux × t = 10 5 4 3 500   = 3 4000 m. 3.12 (B) For minimum number of jumps, range must be maximum. maximum range = g u2 = 10 ) 10 ( 2 = 1 meter. . Total distance to be covered = 10 meter So minimum number of jumps = 10 3.13 (D) y = bx2 dt dy = 2bx. dt dx  dt y d2 = 2 dt dx b 2       + 2 2 dt x d bx 2 a = 2bv2 + 0 v = b 2 a 3.14 (D) Applying equation of motion perpendicular to the incline for y = 0. y = uy t + 2 1 ay t2
• 4. 0 = V sin ( )t + 2 1 (– g cos) t2  t = 0 &     cos g ) sin( V 2 At the moment of striking the plane, as velocity is perpendicular to the inclined plane hence component of velocity along incline must be zero. Vx = ux + ax t 0 = v cos ( – ) + ( – g sin).     cos g ) sin( V 2 v cos ( – ) = tan. 2V sin ( – ) cot ( – ) = 2 tan Ans. (D) 3.15 (A, B, C, D) 0 = u – g sin  .t u g  gcos gsin (a) t = g sin u 2  = 10 30 sin ) 10 .( 2  = 1 sec. (b) t = 2 . 10 3 . 3 10 . 2 = 3 sec. (c) t =  sin g u =   2 3 10 3 10 = 2 sec. t is less than time of flight (d) t =  sin g u = 2 1 . 10 10 = 2 sec. But it's time of flight is 1sec 3.16 (A), (B), (C) (A) Total displacement is zero hence its average velocity is zero. (B) Displacement is zero. (C) Total distance travelled is 2s and total time taken is 2t. < speed > = taken time total travelled ce tan dis total 02 = u2 – 2gs  s = g 2 u2  2s = u2 /g also 0 = u – g t  t= u/g  2 t = g u 2 <speed> = g u 2 g u2 = 2 u
• 5. 3.17 (A), (B), (C) On the curve y = x2 at x = 1/2 y = 4 1 Hence the coordinate       4 1 , 2 1 Differentiating : y = x2  vy = 2xvx vy =       2 1 2 (4) = 4 m/s Which satisfies the line 4x – 4y – 1 = 0 (tangent to the curve) & magnitude of velocity : 2 y 2 x v v | v |    = 2 4 m/s As the line 4x – 4y = 1 does not pass through the origin, therefore (D) is not correct. 3.18 (A, B, D) Let ux and uy be horizontal and vertical components of velocity respectively at t = 0. Then, vy = uy – gt Hence, vy – t graph is straight line. x = vx t Hence, x - t graph is straight line passing through origin. The relation between y and t is y = uy t – ½ gt2 Hence y-t graph is parabolic. vx = constant Hence, vx -t graph is a straight line. 3.19 (A, C, D) R1 =      2 2 cos g ) ( cos sin u 2 and h1 =   cos g 2 sin u 2 2 R2 =      2 2 cos g ) ( cos sin u 2 and h2 =   cos g 2 sin u 2 2 hence h1 = h2 R2 – R1 = g sin  T2 2 R2 – R1 = g sin  T1 2 3.20 (B, C) Total time taken by the ball to reach at bottom = 10 80 x 2 g H 2  = 4 sec. Let time taken in one collision is t Then t x 10 = 7 t = 0.7 sec. No. of collisions = 7 40 = 7 5 5 (5th collisions from wall B)
• 6. Horizontal distance travelled in between 2 successive collisions = 7 m  Horizontal distance travelled in 5/7 part of collisions = 7 x 7 5 = 5 m Distance from A is 2 m. Ans. 3.21 (D) Both the stones cannot meet (collide) because their horizontal component of velocities are different. Hence statement I is false. 3.22 (A) If particle moves with constant acceleration a  , then change in velocity in every one second is numerically equal to a  by definition. Hence statement-2 is true and correct explanation of statement-1. 3.23 (D) Velocity of a particle is independent of its position vector rather it depends on change in position vector while position vector depends on choice of origin. Hence statement-1 is false. 3.24 (B) The question can be reframed as shown in figure. The path of particle is parabolic.  v a    at maximum height, that is at half time of flight u=20m/s =37° a=10m/s2 x y Hence t0 = 10 5 / 3 20 a sin u    = 1.2 sec. 3.25 (B) Speed is least at maximum height, that is at instant to =1.2 sec. 3.26 (C) acceleration and displacement are mutually perpendicular at instant 2t0 = 2.4 sec. 3.27 (A) HA = HC > HB Obviously A just reaches its maximum height and C has crossed its maximum height which is equal to A as u and  are same. But B is unable to reach its max. height. 3.28 (C) Time of flight of A is 4 seconds which is same as the time of flight if wall was not there. Time taken by B to reach the inclined roof is 1 sec. P O Q R TOR = 4 TQR = 1  TOQ = TOR – TQR = 3 seconds. 3.29 (C) From above T = g sin u 2  = 4 s  u sin  = 20 m/s  vertical component is 20 m/s for maximum height v2 = u2 + 2as  02 = 202 – 2 × 10 × s s = 20 m.
• 7. 3.30 (A) r (B) p (C) s (D) q (A) R = m 3 5 ) 10 ( 2 3 100 g 2 sin u2    (B) 11.25 = – 10sin 60º t + 2 1 (10) t2  5t2 – 5 3 t – 11.25 = 0 t = 10 ) 25 . 11 )( 5 ( 4 ) 3 ( 25 3 5   = 10 ) 10 ( 3 3 5  = 3 2 3 3 10 15  R = (10 cos 60)       3 2 3 = 7.5 3 m g cos 30º g g sin 30º 30º 30º (C) t = 3 2 2 3 10 2 1 ) 10 ( 2 º 30 cos g º 30 sin u 2                 sec. R = 10 cos 30º t 2 1  g sin 30º t2 = 3 4 2 1 ) 10 ( 2 1 3 2 2 3 10                = 10 3 20 3 10   m (D) T = 3 4 2 3 10 ) 10 ( 2 30 cos g ) 10 ( 2           sec. R = 2 1 g sin 30º t2 = 2 1 (10)       2 1 3 16 = 3 40 m 3.31 (A) p, q, r (B) p (C) q, t (D) r, s Range of the ball in absence of the wall = g 2 sin u2  = 10 150 sin 202  m = 20 m When d < 20 m, ball will hit the wall. when d = 25 m, ball will fall 5m short of the wall. When d < 20 m, the ball will hit the ground , at a distance, x = 20 m – d in front of the wall.
• 8. 3.32 (4) From graph (1) : vy = 0 at t = 2 1 sec. i.e., time taken to reach maximum height H is t = g uy = 2 1  uy = 5 m/s ...Ans.(i) from graph (2) : vy = 0 at x = 2m i.e., when the particle is at maximum height, its displacement along horizontal x = 2 m x = ux × t  2 = ux × 2 1  ux = 4 m/s ....Ans (ii) 3.33 (3) (a) Taking motion in vertical direction u = 0, g = 10 m/s2 , h = 45 m h = ut + ½ gt2  h = 0 + ½ gt2  t = 10 45 2 g h 2    t = 3 sec. 3.34 (25) Let t be the time after which projectile reaches the ground. Taking motion in horizontal direction 400 = (v0 cos 37°) t 400 = v0 (4/5) t  v0 t = 500 .... (1) Taking motion in vertical direction h = ut + ½ gt2  100 = (– v0 sin 37°) t + ½ (10) t2  100 = 5 3  ( v0 t) + 5 t2 Putting v0 t = 500 from equation (1) ;  100 = 5 3  (500) + 5 t2  5t2 = 400  t = 5 20 sec From eqn (1) ; v0 = 500 × 20 5 v0 = 5 25 m/s
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