Get Solution of These Packages & Learn by Video Tutorials on PROJECTILE MOTION

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1 FREE Download Study Packae from website: & Get Solution of These Packaes & Learn by Video Tutorials on BASIC CONCEPT :. PROJECTILE PROJECTILE MOTION Any object that is ien an initial elocity obliquely and that subsequently follows a path determined by the raitational force actin on it, is called a Projectile. A projectile may be a football, a cricket ball, or any other object.. TRAJECTORY The path followed by a particle (here projectile) durin its motion is called its Trajectory. NOTE :. We shall consider only trajectories that are of sufficiently short rane so that the raitational force can be considered constant in both manitude and direction.. All effects of air resistance will be inored; thus our results are precise only for motion in a acuum on flat, non rotatin Earth..3 PROJECTILE MOTION (i) The motion of projectile is known as projectile motion. (ii) It is an example of two dimensional motion with constant acceleration. (iii) Projectile motion is considered as combination of two simultaneous motions in mutually perpendicular directions which are completely independent from each other i.e. horizontal motion and ertical motion. + Parabolic path ertical motion + horizontal motion.. PROJECTILE THROWN AT AN ANGLE WITH HORIZONTAL (i) (ii) Consider a projectile thrown with a elocity u makin an anle θ with the horizontal. Initial elocity u is resoled in components in a coordinate system in which horizontal direction is taken as x-axis, ertical direction as y-axis and point of projection as oriin. Teko Classes, Maths : Suha R. Kariya (S. R. K. Sir), Bhopal Phone : , pae

2 FREE Download Study Packae from website: & Get Solution of These Packaes & Learn by Video Tutorials on (iii) u x u cos θ u y u sin θ Aain this projectile motion can be considered as the combination of horizontal and ertical motion. Therefore, Horizontal direction Vertical direction (a) Initial elocity u x u cos θ Initial elocity u y u sin θ (b) Acceleration a x 0 Acceleration a y (c) Velocity after time t, x u cos θ Velocity after time t, u sin θ t. TRAJECTORY EQUATION : If we consider the horizontal direction, x u x.t For ertical direction : x u cos θ. t...() y u y. t / t Substitutin the alue x equation () u sin θ. t / t...() y u sin θ. x ucosθ x ucosθ This is an equation of parabola called as trajectory equation of projectile motion.. TIME OF FLIGHT : Since the displacement alon ertical direction does not occur. So, Net displacement 0 (u sin θ) T T 0 T usin θ.3 HORIZONTAL RANGE : R u x.t R u cos θ. u sinθ R usin θ.4 MAXIMUM HEIGHT : Usin 3 rd equation of motion i.e. u + as we hae for ertical direction 0 u sin θ H H u sin θ Teko Classes, Maths : Suha R. Kariya (S. R. K. Sir), Bhopal Phone : , pae

3 Get Solution of These Packaes & Learn by Video Tutorials on FREE Download Study Packae from website: & NOTE:.5 RESULTANT VELOCITY : x î + y ĵ u cos θ î + (u sin θ t) ĵ where u cos θ + (usinθ t) and tan θ / x. Results of article.,.4 and.4 are alid only for complete fliht, that is when proejctile lands at same horizontal leel from which it has been projected. Vertical component of elocity is zero when particle moes horizontally, i.e., at the hihest point of trajectory. Vertical component of elocity is positie when particle is moin up and ertical component of elocity is neatie when particle is comin down if ertical upwards direction is taken as positie. Any direction upward or downward can be taken as positie and if downward direction is taken as positie then ertical component of elocity comin down is positie..6 GENERAL RESULT : (i) For maximum rane θ 45º (ii) u R max In this situation H max R max We et the same rane for two anle of projections α and (90 α) but in both cases, maximum heihts attained by the particles are different. (iii) Ιf R H (i) i.e. u sinθ tan θ 4 Rane can also be expressed as u sinθ R u xuy () CHANGE IN MOMENTUM : (a) (b) u sin θ usin θ.ucos θ Initial elocity u i u cos θ î + u sin θ ĵ Final elocity u f u cos θ î u sin θ ĵ Chane in elocity for complete motion Teko Classes, Maths : Suha R. Kariya (S. R. K. Sir), Bhopal Phone : , pae 3

4 FREE Download Study Packae from website: & Get Solution of These Packaes & Learn by Video Tutorials on Ex. (c) (d) u u f u u sin θ ĵ i Chane in momentum for complete motion P P f Pi m ( u f ui ) m( u sin θ) ĵ mu sin θ ĵ Velocity at the hihest point u f u cos θ î Chane in momentum at hihest point m ( u f u ) m [u cos θ î (u cos θ î + u sin θ ĵ ) mu sin θ ĵ i A body is projected with a speed of 30 ms at an anle of 30º with the ertical. Find the maximum heiht, time of fliht and the horizontal rane of the motion. [ Take m/s ] Sol. Here u 30 ms, Anle of projection, θ º Ex. Sol. u sin θ 30 sin 60º Maximum heiht, H Time of fliht, T usin θ 30 sin sec. 35 m 4 A stone is thrown with a elocity at anle θ with horizontal. Find its speed when it makes an anle β with the horizontal. Since horizontal component of elocity remains constant. Therefore, cosθ u cos β u cosθ cosβ Ex.3 A projectile is thrown in the upward direction makin an anle of 60 0 with the horizontal with a speed of 47 m/s. Find the time after which its inclination with the horizontal is 45 0? Sol. u x 47 cos u y 47 sin u y + a y t t x u x 47 When anle is 45 0, tan 45 0 x y x t Teko Classes, Maths : Suha R. Kariya (S. R. K. Sir), Bhopal Phone : , pae 4

5 Get Solution of These Packaes & Learn by Video Tutorials on FREE Download Study Packae from website: & Ex.4 Sol ( 3 ) t t ( 3 ) s A lare number of bullets are fired in all directions with the same speed. What is the maximum area on the round on which these bullets will spread? Maximum distance upto which a bullet can be fired is its maximum rane, therefore R max 4 π Maximum area π(r max ). Ex.5 The elocity of projection of a projectile is ien by : u 5î + ĵ. Find (a) Time of fliht, (b) Maximum heiht, (c) Rane Sol. We hae u x 5 u y Ex.6 Sol. (a) Time of fliht (b) Maximum heiht usin θ u y u sin θ usin θ.ucos θ (c) Rane HEIGHT AND RANGE : u y s 5 m 5 m A batter hits a baseball so that it leaes the bat with an initial speed m/s at an initial anle α 0 53º, at a location where.0 m/s (a) Find the position of the ball, and the manitude and direction of its elocity, when t.0 s. (b) (c) Find the time when the ball, reaches the hihest point of fliht and the find its heiht h at this point Find the horizontal rane R - that is, the horizontal distance from the startin point to the point at which the ball hits the round. For each part, treat the baseball as a projectile The initial elocity of the ball has components (a) 0x 0 cos α 0 (37.0 m/s) cos 53º.3 m/s 0y 0 sin α 0 (37.0 m/s) sin 53º 9.5 m/s x 0x t (.3 m/s) (.00 s) 44.6 m y 0y t t (9.5 m/s) ( s) ( m/s ) ( s) m x 0x.3 m/s y x 46 m t.00 s t 3 s m/s α 0 53º y 40 m h 45 m t 6 s R 38 m x Teko Classes, Maths : Suha R. Kariya (S. R. K. Sir), Bhopal Phone : , pae 5

6 FREE Download Study Packae from website: & Get Solution of These Packaes & Learn by Video Tutorials on Ex.7 0y t 9.5 m/s ( m/s ) (.00 s) 9.5 m/s The y-component of elocity is positie, which means that the ball is still moin upward at this time (Fiure). The manitude and direction of the elocity are (b) x + (.3m/ s) + (9.5m/ s) 4. m/s.0m / s α tan.3m/ s tan 0.4 At the hihest point, the ertical elocity is zero at time t ; then 0 0y t t 0 y 9.5 m/s m/s 3.0 s The heiht h at this time is the alue of y when t t 3 s; h 0y t t (9.5 m/s) (3.0 s) (.0 m/s ) (3.0 s) m (c) y 0 0y t t t 0y t This is a quadratic equation for t. It has two roots, 0y t 0 and t (9.5 m/ s) 5.9 s m/ s There are two times at which y 0; t 0 is the time the ball leaes the round, and t 6 s is the time of its return. This is exactly twice the time to reach the hihest point, so the time of descent equals the time of ascent. (This is always true if the startin and end points are at the same eleation and air resistance can be nelected). The horizontal rane R is the alue of x when the ball returns to the round, that is, at t 6.64 s; R 0x t (.3 m/s) (5.9 s) 3.6 m The ertical component of elocity when the ball hits the round is 0y t 9.5 m/s ( m/s ) (5.9 s) 9.5 m That is, has the same manitude as the initial ertical elocity 0y but the opposite direction (down). Since x is constant, the anle α 53º )(below the horizontal) at the point is the neatie of the initial anle α 0 53º. A cleer monkey escapes from the zoo. The zoo keeper finds him in a tree. After failin to entice the monkey and shoots. The cleer monkey lets o at the same instant the dart leaes the un barrel, intendin to land on the round and escape. Show that the dart always hits the monkey, reardless of the dart s muzzle elocity (proided that it ets to the monkey before he hits the round). Teko Classes, Maths : Suha R. Kariya (S. R. K. Sir), Bhopal Phone : , pae 6

7 Get Solution of These Packaes & Learn by Video Tutorials on FREE Download Study Packae from website: & Sol. The monkey drops straiht down, so x monkey d at all times. For the dart, x dart ( 0 cos a 0 )t. When these x- coordinates are equal, d ( 0 cos α 0 )t, or Ex.8 t d 0 cos α 0 To hae the dart hit the monkey, it must be true that y monkey y dart at this same time. The monkey is in onedimensional free fall y monkey d tan α 0 t. For the dart y dart ( 0 sin α 0 ) t t So if d tan α 0 ( 0 sin α 0 ) t at the time when the two x-coordinates are equal, then y monkey y dart and we hae a hit. A ball is thrwon from round leel so as to just clear a wall 4 m hih at a distance of 4 m and falls at a distance of 4 m from the wall. Find the manitude and direction of the ball. Sol. The ball passes throuh the point P(4, 4). So its rane m. The trajectory of the the ball is, Now y x tan θ ( R x ) x 4m, y 4m and R 8 m tan θ 4 tan θ. 8 9 or tanθ 7 9 And R or 8 u sinθcosθ u θ tan or u or u 8 and θ tan. 7 Teko Classes, Maths : Suha R. Kariya (S. R. K. Sir), Bhopal Phone : , pae 7

8 FREE Download Study Packae from website: & Get Solution of These Packaes & Learn by Video Tutorials on Qus. Ans. Two projectiles are thrown with different speeds and at different anles so as the coer the same maximum heiht. Find out the sum of the times taken by each to the reach to hihest point, if time of fliht is T. Total time taken by either of the projectile. Qus. A particle is projected with speed m/s at an anle 60 0 with horizontal. Find : (a) time of fliht (b) rane (c) maximum heiht (d) elocity of particle after one second. (e) elocity when heiht of the particle is m 5 Ans. (a) 3 sec. (b) 5 3 m (c) m (d) 5.7 m/s 4 (e) 5î ± 55 ĵ 3. PROJECTILE THROWN PARALLEL TO T THE HORIZONTAL : Consider a projectile thrown from point O at some heiht h from the round with a elocity u. Now we shall deal the characteristics of projectile motion with the help of horizontal and ertical direction motions. Horizontal direction Vertical direction (i) Initial elocity u x u Initial elocity u y 0 (ii) Acceleration a x 0 Acceleration a y (downward) 3. TRAJECTORY EQUATION : The path traced by projectile is called the trajectory. After time t, From equation () Put alue of t in equation () x ut...() y t...() x y. u t x/u This is trajectory equation of the projectile. 3. Velocity at a eneral point P(x, y) u x + u y Here horizontal elocity of the projectile after time t x u elocity of projectile in ertical direction after time t 0 + ( )t t t (downward) Teko Classes, Maths : Suha R. Kariya (S. R. K. Sir), Bhopal Phone : , pae 8

9 Get Solution of These Packaes & Learn by Video Tutorials on u + t and tan θ / x FREE Download Study Packae from website: & Ex.9 Sol. 3.3 DISPLACEMENT : The displacement of the particle is expressed by S x î + y ĵ (ut) î + ( t ) ĵ where S x + y 3.4 TIME OF FLIGHT : This is equal to the time taken by the projectile to return to round. From equation of motion S ut + at Therefore for ertical direction h t + ( )t At hihest point 0 h t t ± h t 3.5 HORIZONTAL RANGE : Distance coered by the projectile alon the horizontal direction between the point of projection to the point on the round. R u x. t R u h 3.6 VELOCITY AT VERTICAL DEPTH h : Alon ertical direction h 0 +. ( h) ( ) EXAMPLES BASED ON PROJECTILE PROJECTED HORIZONTALLY A projectile is fired horizontally with a speed of 98 ms from the top of a hill 490 m hih. Find (i) the time taken to reach the round (ii) the distance of the taret from the hill and (iii) the elocity with which the projectile hits the round. (take 9.8 m/s ) (i) The projectile is fired from the top O of a hill with speed u 98 ms alon the horizontal as shown as OX. It reaches the taret P at ertical depth OA, in the coordinate system as shown, OA y 490 m h Teko Classes, Maths : Suha R. Kariya (S. R. K. Sir), Bhopal Phone : , pae 9

10 Get Solution of These Packaes & Learn by Video Tutorials on FREE Download Study Packae from website: & Ex. Sol. As, y t t or t 0 s. (ii) (iii) Distance of the taret from the hill is ien by, AP x Horizontal elocity time m. The horizontal and ertical components of elocity of the projectile at point P are x u 98 ms u y + t ms V x ms Now if the resultant elocity makes air anle β with the horizontal, then tan β y u 98 β 45º 98 A motorcycle stunt rider rides off the ede of a cliff. Just at the ede his elocity is horizontal, with manitude 9.0 m/s. Find the motorcycle s position, distance from the ede of the cliff and elocity after 0.5 s. At t 0.50 s, the x and y-coordinates are x 0x t (9.0 m/s) (0.50 s) 4.5 m y t ( m/s ) (0.50 s). m The neatie alue of y shows that a this time the motorcycle is below its startin point. The motorcycle s distance from the oriin at this time r x + y ( 4.5m) + (.m) The components of elocity at this time are 45 m + m 3 (5) + (4) ~ 5 sec. Teko Classes, Maths : Suha R. Kariya (S. R. K. Sir), Bhopal Phone : , pae

11 FREE Download Study Packae from website: & Get Solution of These Packaes & Learn by Video Tutorials on Qus. Ans. Qus. Ans. x 0x 9.0 m/s t ( m/s ) (0.50 s) 5 m/s. The speed (manitude of the elocity) at this time is x + ( 9.0m/ s) + ( 5m/ s). m/s The anle α of the elocity ector is tan y x 5 m/ s tan 9.0 m/ s Two tall buildins face each other and are at a distance of 80 m from each other. With what elocity must a ball be thrown horizontally from a window 55 m aboe the round in one buildin, so that it enters a window.9 m aboe the round in the second buildin. 60 m/s. Two paper screens A and B are separated by a distance of 0 m. A bullet pierces A and then B. The hole in B is cm below the hole in A. If the bullet is traellin horizontally at the time of hittin the screen A, calculate the elocity of the bullet when it hits the screen A. Nelect the resistance of paper and air. 700 m/s 4. PROJECTILE FROM A TOWER Case (i) : Case (ii) : Case (iii) : Ex. Horizontal projection u x u ; u y 0 ; a y Projection at an anle θ aboe horizontal u x ucosθ ; u y usinθ; a y Projection at an anle θ below horizontal u x ucosθ; u y usinθ; a y In all the aboe three cases, we can calculate the elocity of projectile at the instant of strikin the round by usin x + y and tanφ y x, where φ is the anle at which the projectile strikes the round. From the top of a 50m hih tower a stone is projected with speed m/s, at an anle of 37º as shown in fiure. Find out (a) elocity after 3s (b) time of fliht. (c) horizontal rane. (d) the maximum heiht attained by the particle. Sol. (a) Initial elocity in horizontal direction cos 37 8 m/s Initial elocity in ertical direction sin 37º 6 m/s Velocity after 3 seconds Teko Classes, Maths : Suha R. Kariya (S. R. K. Sir), Bhopal Phone : , pae

12 Get Solution of These Packaes & Learn by Video Tutorials on x î + ĵ FREE Download Study Packae from website: & Qus. 8 î + (u y + a y t) ĵ 8 î 4 ĵ (b) S y u y t + ay t 50 6 t + ( ) t 5t 6t 50 0 t 6 ± (c) Rane 8 ( 36 (d) u y + a y t 0 6 t t 0.6 or 0 6 h h.8 maximum heiht m. ) 6 ± 36 Two stones A and B are projected simultaneously from the top of a 0 m hih tower. Stone B is projected horizontally with speed m/s, and stone A is dropped from the tower. Find out the followin (a) time of fliht of the two stone. (b) distance between two stones after 3 sec. (c) anle of strike with round. (d) horizontal rane of particle B. Ans. (a) 5 sec. (b) x B 30 m, y B 45 (c) tan 5 (d) 0 5 m 5. PROJECTION FROM A MOVING BODY Case (i) : Case (ii) : Case (iii) : Case (i) : Consider a boy standin on a trolley who throws a ball with speed u at an anle θ with the horizontal. Trolley moes horizontally with constant speed. When ball is projected in the direction of motion of the trolley, horizontal component of ball s elocity u cos θ + Initial ertical component of ball s elocity u sin θ The ball is projected opposite to the direction of motion of the trolley Horizontal component of ball s elocity u cosθ Initial ertical component of ball s elocity u sinθ The ball projected upwards from a platform moin with speed upwards. Horizontal component of ball s elocity u cos θ Initial ertical component of ball s elocity u sin θ + The ball projected upwards from a platform moin with speed downwards. Horizontal component of ball s elocity u cosθ Initial ertical component of ball s elocity u sinθ Ex. A particle is projected at an anle of 30 0 with speed 0 m/s : (i) Find out position ector of the particle after s (ii) Find out anle betwen elocity ector and position ector at t s Sol. (i) s x u cos θ t Teko Classes, Maths : Suha R. Kariya (S. R. K. Sir), Bhopal Phone : , pae

13 Get Solution of These Packaes & Learn by Video Tutorials on FREE Download Study Packae from website: & Ex.3 (ii) 0 3 t 3 m s y u sin θt + t 0 5 () 5m Position ector 3 î + 5 ĵ x 3 î u y + a y t 0 3 î. s s cos θ cos θ θ cos A boy standin on a lon railroad car throws a ball straiht upwards. The car is moin on the horizontal road with an acceleration of m/s and the projection speed in the ertical direction is 9.8 m/s. How far behind the boy will the ball fall on the car? Sol. Let the initial elocity of car be u. t Qus. where u u component of elocity in ertical direction x c u + u + where x c distance traelled by car x b distance traelled by ball x b u x c x b u + u m Ans. A person is standin on a truck moin with a constant elocity of 4.7 m/s on a horizontal road. The man throws a ball in such a way that it returns to the truck after the truck has moed 58.8m. Find the speed and the anle of projection (a) as seen from the truck, (b) as seen from the road. [Ans : (a) 9.6 m/s upward (b) 4.5 m/s at 53º with horizontal] 6. PROJECTION ON AN INCLINED PLANE Case (i) : To sole the problem of projectile motion on an incline plane we can adopt two types of axis system as shown in the fiures Up the incline 3 3 Teko Classes, Maths : Suha R. Kariya (S. R. K. Sir), Bhopal Phone : , pae 3

14 Get Solution of These Packaes & Learn by Video Tutorials on FREE Download Study Packae from website: & Case (ii) : axis system axis system Here α is anle of projection Here α is anle of projection with the horizontal. with the inclined plane In this case: In this case: a x 0 u x ucosα a x - sinβ u x ucosα a y u y usinα a y - cosβ u y usinα Time of fliht (T) : Down the incline when the particle strikes the inclined plane y coordinate becomes zero y u y t + ay t 0 usinαt cosβt usin α T cosβ u Maximum heiht (H) : when half of the time is elasped y coordinate is equal to maximum heiht of the projectile usinα H u sinα cosβ sin u u sin α H cos β usinα β cos β Rane alon the inclined plane (R): When the particle strikes the inclined plane x coordinate is equal to rane of the particle x u x t + ax t usin α R ucosθ cosβ R u sinαcos( α + β) cos β cos usinα β cos β Teko Classes, Maths : Suha R. Kariya (S. R. K. Sir), Bhopal Phone : , pae 4

15 Get Solution of These Packaes & Learn by Video Tutorials on FREE Download Study Packae from website: & axis system axis system In this case : In this case : a x 0 u x ucosα a x sinβ u x ucosα a y u y usinα a y cosβ u y usinα Time of fliht (T) : when the particle strikes the inclined plane y coordinate becomes zero y u y t + ay t 0 usinαt cosβt usin α T cosβ Maximum heiht (H) : u when half of the time is elasped y coordinate is equal to maximum heiht of the projectile usinα H u sinα cosβ sin u sin α H cosβ u usinα β cos β Rane alon the inclined plane (R): When the particle strikes the inclined plane x coordinate is equal to rane of the particle x u x t + ax t usin α R ucosθ cosβ + R u sinαcos( α β) cos Table : Standard results for projectile motion on an inclined plane β cos usin α β cos β Teko Classes, Maths : Suha R. Kariya (S. R. K. Sir), Bhopal Phone : , pae 5

16 FREE Download Study Packae from website: & Get Solution of These Packaes & Learn by Video Tutorials on Rane Time of fliht u Up the Incline sin α cos( α + β) cos β u sin α cosβ Anle of projection for π β maximum rane 4 Maximum Rane u Down the Incline u sinα cos( α β) cos β usin α cosβ π β + 4 ( + sinβ) ( sinβ) Here α is the anle of projection with the incline and β is the anle of incline. NOTE: For a ien speed, the direction - which ies the maximum rane of the projectile on an incline, bisects the anle between the incline and the ertical, for upward or downward projection. Ex.4 Sol. A particle is projected horizontally with a speed u from the top of a plane inclined at an anle θ with the horizontal. How far from the point of projection will the particle strike the plane? Take X,Y-axes as shown in fiure. Suppose that the particle strikes the plane at a point P with coordinates (x,y). Consider the motion between A and P. Motion in x direction : Initial elocity u Acceleration 0 x ut...(i) Motion in y direction : Initial elocity u Acceleration Eliminatin t from (i) and (ii) y y t...(ii) x u Also y xtan θ x Thus, xtan θ iin x 0 or, u u tan θ u tan θ Clearly the point P corresponds to x then y xtan θ The distance AP u tan x + y θ u Teko Classes, Maths : Suha R. Kariya (S. R. K. Sir), Bhopal Phone : , pae 6

17 Get Solution of These Packaes & Learn by Video Tutorials on FREE Download Study Packae from website: & u tan θ + tan θ u tan θ sec θ Ex.5 A projectile is thrown at an anle θ with an inclined plane of inclination β as shown in fiure. Find the relation between β and θ if : (a) projectile strikes the inclined plane perpendicularly, (b) projectile strikes the inclined plane horizontal. Sol. (a) If projectile strikes perpendicularly. x 0 when projectile strikes x u x + a x t 0 u cos θ sin β T (b) T we also know that T ucosθ sinβ ucosθ sinβ usin θ cosβ usin θ cosβ tan θ cot β If projectile strikes horizontally, then at the time of strikin the projectile will be at the maximum heiht from the round. Therefore : t 0P t 0P usin θ cosβ usin( θ + β) usin θ cosβ usin( θ + β) sin θ sin (θ + β) cos β. Elastic collision of a projectile with a wall : Suppose a projectile is projected with speed u at an anle θ from point O on the round. Rane of the projectile is R. If a wall is present in the path of the projectile at a distance x from the point O. The collision with the wall is elastic, path of the projectile chanes after the collision as described below. Case I : If x R Direction of x component of elocity is reersed but its manitude remains the same and y component of elocity remains unchaned, therefore the remainin distance (R x) is coered in the backward direction and projectile falls a distance (R x) ahead of the point O as shown in fiure. Case II : If x < R Direction of x component of elocity is reersed but its manitude remains the same and y component of elocity remains unchaned, therefore the remainin distance (R x) is coered in the backward direction and projectile falls a distance (R x) behind the the point O as shown in fiure. Teko Classes, Maths : Suha R. Kariya (S. R. K. Sir), Bhopal Phone : , pae 7