Motion in Two Dimension (Projectile Motion)

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1 Phsics Motion in Two Dimension (Projectile Motion)

2 Table of Content. Introdction.. Projectile. 3. Assmptions of projectile motion. 4. Principle of phsical independence of motions. 5. Tpes of projectile motion. 6. bliqe projectile. 7. orizontal projectile. 8. Projectile on an inclined plane.

3 The motion of an object is called two dimensional, if two of the three co-ordinates are reqired to specif the position of the object in space chanes w.r.t time. In sch a motion, the object moves in a plane. For eample, a billiard ball movin over the billiard table, an insect crawlin over the floor of a room, earth revolvin arond the sn etc. Two special cases of motion in two dimension are:. Projectile motion. Circlar motion PRJECTILE MTIN. Introdction. A hnter aims his n and fires a bllet directl towards a monke sittin on a distant tree. If the monke remains in his position, he will be safe bt at the instant the bllet leaves the barrel of n, if the monke drops from the tree, the bllet will hit the monke becase the bllet will not follow the linear path. The path of motion of a bllet will be parabolic and this motion of bllet is defined as projectile motion. If the force actin on a particle is obliqe with initial velocit then the motion of particle is called projectile motion.

4 . Projectile. A bod which is in fliht throh the atmosphere bt is not bein propelled b an fel is called projectile. Eample: (i) A bomb released from an airplane in level fliht (ii) A bllet fired from a n (iii) An arrow released from bow (iv) A Javelin thrown b an athlete 3. Assmptions of Projectile Motion. () There is no resistance de to air. () The effect de to crvatre of earth is neliible. (3) The effect de to rotation of earth is neliible. (4) For all points of the trajector, the acceleration de to ravit is constant in manitde and direction. 4. Principles of Phsical Independence of Motions. () The motion of a projectile is a two-dimensional motion. So, it can be discssed in two parts. orizontal motion and vertical motion. These two motions take place independent of each other. This is called the principle of phsical independence of motions. () The velocit of the particle can be resolved into two mtall perpendiclar components. orizontal component and vertical component. (3) The horizontal component remains nchaned throhot the fliht. The force of ravit continosl affects the vertical component. (4) The horizontal motion is a niform motion and the vertical motion is a niforml accelerated retarded motion. 3

5 5. Tpes of Projectile Motion. () bliqe projectile motion () orizontal projectile motion (3) Projectile motion on an inclined plane 6. bliqe Projectile. In projectile motion, horizontal component of velocit ( cos), acceleration () and mechanical ener remains constant while, speed, velocit, vertical component of velocit ( sin ), momentm, kinetic ener and potential ener all chanes. Velocit, and KE are maimm at the point of projection while minimm (bt not zero) at hihest point. () Eqation of trajector: A projectile thrown with velocit at an anle with the horizontal. The velocit can be resolved into two rectanlar components. v cos component alon ais and sin component alon ais. For horizontal motion For vertical motion From eqation (i) and (ii) tan = cos t t. (i) cos ( sin ) t t. (ii) sin cos cos This eqation shows that the trajector of projectile is parabolic becase it is similar to eqation of parabola = a b cos P sin cos 4

6 Note: Eqation of obliqe projectile also can be written as tan R (Where R = horizontal rane = sin ) () Displacement of projectile ( r ) : Let the particle acqires a position P havin the coordinates (, ) jst after time t from the instant of projection. The correspondin position vector of the particle at time t is r as shown in the fire. r i ˆ j ˆ.(i) The horizontal distance covered drin time t is iven as v t cos t.(ii) The vertical velocit of the particle at time t is iven as v ( v 0 ) t,.(iii) Now the vertical displacement is iven as sin t / t.(iv) Pttin the vales of and from eqation (ii) and eqation (iv) in eqation (i) we obtain the position vector at an time t as r tˆ ( cos) i ( sin ) t t ˆj r ( t cos ) ( t sin ) t P (, ) vi t tsin r t and tan ( / ) sin tan t cos tan t sin / t ( t cos) or Note: The anle of elevation of the hihest point of the projectile and the anle of projection are related to each other as tan tan R 5

7 (3) Instantaneos velocit v: In projectile motion, vertical component of velocit chanes bt horizontal component of velocit remains alwas constant. Eample: When a man jmps over the hrdle leavin behind its skateboard then vertical component of his velocit is chanin, bt not the horizontal component, which matches with the skateboard velocit. As a reslt, the skateboard stas nderneath him, allowin him to land on it. Let v i be the instantaneos velocit of projectile at time t direction of this velocit is alon the tanent to the trajector at point P. v v i v ˆj v i i v v cos ( sin t) v i t tsin Direction of instantaneos velocit v sin t tan or tan t v cos tan sec (4) Chane in velocit: Initial velocit (at projection point) cos ˆi sin ˆj i Final velocit (at hihest point) f cos ˆi 0 ˆj (i) Chane in velocit (Between projection point and hihest point) f i sin ˆj When bod reaches the rond after completin its motion then final velocit (ii) Chane in velocit (Between complete projectile motions) f i f sin ˆi cos ˆi sin ˆj (5) Chane in momentm: Simpl b the mltiplication of mass in the above epression of velocit (Article-4). (i) Chane in momentm (Between projection point and hihest point) p p f p i msin ˆj (ii) Chane in momentm (For the complete projectile motion) p p f p i msin ˆj 6

8 (6) Anlar momentm: Anlar momentm of projectile at hihest point of trajector abot the point of projection is iven b L mvr ere r L m cos sin m 3 cos sin sin P = mv r (7) Time of fliht : The total time taken b the projectile to o p and come down to the same level from which it was projected is called time of fliht. For vertical pward motion 0 = sin t t = ( sin /) Now as time taken to o p is eqal to the time taken to come down so Time of fliht sin T t (i) Time of fliht can also be epressed as: velocit). (ii) For complementar anles of projection and 90 o T (a) Ratio of time of fliht = T (b) Mltiplication of time of fliht =. T (where is the vertical component of initial sin / T = tan tan sin(90 )/ T sin cos TT T T (iii) If t is the time taken b projectile to rise p to point p and t is the time taken in fallin from point sin p to rond level then t t time of fliht or ( t ) sin t and heiht of the point p is iven b b solvin ( t h t t t h ) t t h sin t (iv) If B and C are at the same level on trajector and the time difference between these two points is t, similarl A and D are also at the same level and the time difference between these two positions is t then t R t P h t 7

9 t t 8h h B A t t C D (8) orizontal rane: It is the horizontal distance travelled b a bod drin the time of fliht. So b sin second eqation of motion R cos T cos ( sin / ) R sin (i) Rane of projectile can also be epressed as: sin sin cos sin R = cos T = cos velocit) R (Where and are the horizontal and vertical component of initial orizontal rane (ii) If anle of projection is chaned from to = (90 ) then rane remains nchaned. 60 o 30 o Blast R' o sin ' sin[(90 )] sin So a projectile has same rane at anles of projection and (90 ), thoh time of fliht, maimm heiht and trajectories are different. These anles and 90 o are called complementar anles of projection and for complementar R R sin / R anles of projection ratio of rane o R sin[(90 )]/ R 8

10 Ma. heiht (iii) For anle of projection = (45 ) and = (45 + ), rane will be same and eqal to cos /. and are also the complementar anles. (iv) Maimm rane: For rane to be maimm dr d d sin 0 0 d cos = 0 i.e. = 90 o = 45 o and R ma = ( /) i.e., a projectile will have maimm rane when it is projected at an anle of 45 o to the horizontal and the maimm rane will be ( /). When the rane is maimm, the heiht reached b the projectile Rma sin sin i.e., if a person can throw a projectile to a maimm distance R ma, The maimm heiht to which it will rise is R ma 4. (v) Relation between horizontal rane and maimm heiht: R sin / 4 cot sin / R 4 cot sin R and (vi) If in case of projectile motion rane R is n times the maimm heiht i.e. R = n The anle of projection is iven b tan [4 / n] sin sin n tan [4 / n ] or tan [4 / n] 45 o Rma = 4 sin Note : If R = then tan o (4) or 76. If R = 4 then tan o () or 45. (9) Maimm heiht: It is the maimm heiht from the point of projection, a projectile can reach. So, b sin v as 0 ( sin ) sin (i) Maimm heiht can also be epressed as 9

11 (ii) (where is the vertical component of initial velocit). ma (when sin = ma = i.e., = 90 o ) i.e., for maimm heiht bod shold be projected verticall pward. So it falls back to the point of projection after reachin the maimm heiht. (iii) For complementar anles of projection and 90 o Ratio of maimm heiht = tan sin sin (90 / o ) sin tan cos (0) Projectile passin throh two different points on same heiht at time t and t : If the particle passes two points sitated at eqal heiht at t t and t t, then sin t t...(i) (i) eiht (): sin t t...(ii) and t = t t = t Comparin eqation (i) with eqation (ii) t sin t Sbstittin this vale in eqation (i) t t t t t t (ii) Time (t and t ): sin t t t sin t 0 sin t sin sin t and sin t sin sin 0

12 () Motion of a projectile as observed from another projectile: Sppose two balls A and B are projected simltaneosl from the oriin, with initial velocities and at anle and, respectivel with the horizontal. The instantaneos positions of the two balls are iven b Ball A : = ( cos )t Ball B : = ( cos )t ( sin ) t t ( sin ) t t The position of the ball A with respect to ball B is iven b Now cos cos ) t ( sin sin ) t ( sin cos sin constant cos Ths motion of a projectile relative to another projectile is a straiht line. () Ener of projectile: When a projectile moves pward its kinetic ener decreases, potential ener increases bt the total ener alwas remain constant. If a bod is projected with initial kinetic ener K(=/ m ), with anle of projection with the horizontal then at the hihest point of trajector (i) Kinetic ener K' K cos m ( cos) m cos K B cos K = Kcos (ii) Potential ener m m sin (iii) Total ener = Kinetic ener + Potential ener = sin m sin As m m = Ener at the point of projection. This is in accordance with the law of conservation of ener. cos m sin

13 7. orizontal Projectile. A bod be projected horizontall from a certain heiht verticall above the rond with initial velocit. If friction is considered to be absent, then there is no other horizontal force which can affect the horizontal motion. The horizontal velocit therefore remains constant and so the object covers eqal distance in horizontal direction in eqal intervals of time. () Trajector of horizontal projectile: The horizontal displacement is overned b the eqation = t t. (i) The vertical displacement is overned b (Since initial vertical velocit is zero) B sbstittin the vale of t in eqation (ii) t. (ii) () Displacement of Projectile (r ) : After time t, horizontal displacement t and vertical displacement t. So, the position vector r t i Therefore ˆ t ˆj t t r t and tan tan as (3) Instantaneos velocit: Throhot the motion, the horizontal component of the velocit is v =. t The vertical component of velocit increases with time and is iven b So, i.e. v t Aain v = 0 + t = t (From v = + t) v v ˆi v ˆ j = v ˆi t ˆj v i ˆ ˆj t P (,) v P(, ) v v i.e. v

14 Direction of instantaneos velocit: v tan v v tan v tan or tan t Where is the anle of instantaneos velocit from the horizontal. (4) Time of fliht: If a bod is projected horizontall from a heiht h with velocit and time taken b the bod to reach the rond is T, then h 0 T (For vertical motion) h T (5) orizontal rane: Let R is the horizontal distance travelled b the bod R T 0 T (For horizontal motion) h R (6) If projectiles A and B are projected horizontall with different initial velocit from same heiht and third particle C is dropped from same point then (i) All three particles will take eqal time to reach the rond. (ii) Their net velocit wold be different bt all three particle possess same vertical component of velocit. (iii) The trajector of projectiles A and B will be straiht line w.r.t. particle C. h C A B 3

15 (7) If varios particles thrown with same initial velocit bt indifferent direction then A h A A B C D E E (i) The strike the rond with same speed at different times irrespective of their initial direction of velocities. (ii) Time wold be least for particle E which was thrown verticall downward. (iii) Time wold be maimm for particle A which was thrown verticall pward. 8. Projectile Motion on an Inclined Plane. Let a particle be projected p with a speed from an inclined plane which makes an anle with the horizontal velocit of projection makes an anle with the inclined plane. We have taken reference -ais in the direction of plane. ence the component of initial velocit parallel and perpendiclar to the plane are eqal to and sin respectivel i.e. cos and sin The component of alon the plane is the plane is and a cos.. sin and perpendiclar to cos as shown in the fire i.e. a sin Therefore the particle decelerates at a rate of from to P. () Time of fliht: We know for obliqe projectile motion or we can sa T a Time of fliht on an inclined plane sin T cos sin as it moves sin T t =0 a= sin a= cos cos P t =T 4

16 () Maimm heiht: We know for obliqe projectile motion or we can sa a Maimm heiht on an inclined plane sin cos sin (3) orizontal rane: For one dimensional motion s t at orizontal rane on an inclined plane R T a T R cos T sin T B solvin sin sin R cos sin cos cos sin cos( ) R cos (i) Maimm rane occrs when 4 (ii) The maimm rane alon the inclined plane when the projectile is thrown pwards is iven b R ma ( sin) (iii) The maimm rane alon the inclined plane when the projectile is thrown downwards is iven b R ma ( sin) 5

Ground Rules. PC1221 Fundamentals of Physics I. Position and Displacement. Average Velocity. Lectures 7 and 8 Motion in Two Dimensions

Ground Rules. PC1221 Fundamentals of Physics I. Position and Displacement. Average Velocity. Lectures 7 and 8 Motion in Two Dimensions PC11 Fndamentals of Physics I Lectres 7 and 8 Motion in Two Dimensions A/Prof Tay Sen Chan 1 Grond Rles Switch off yor handphone and paer Switch off yor laptop compter and keep it No talkin while lectre