XI PHYSICS M. AFFAN KHAN LECTURER PHYSICS, AKHSS, K.

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1 XI PHYSICS M. AFFAN KHAN LECTURER PHYSICS, AKHSS, K affan_414@live.com [MOTION IN TWO DIMENSIONS] CHAPTER NO. 4 In this chapter we are oin to discuss motion in projectile motion and circular motion in which we will be learnin Kinematics as well as Dynamics of Circular Motion and will also be learnin some observatory mathematics for problem solvin in Projectile motion.

2 MOTION OF PROJECTILE Projectile Motion A body which is thrown in air at certain anle θ with the horizontal, the body will move only under the action of ravity, called as projectile and since the body will trace a Parabolic path, this type of motion is known as Projectile Motion. Examples: Motion of a ball or football or basketball in air, motion of water drops from fountain, motion of cannon ball etc. Assumptions in Projectile Motion To make the analysis simple for projectile motion followin assumptions are made. Air resistance is nelected Motion of the Earth is nelected The value of ravitational acceleration is constant everywhere Velocity of Projectile Consider a projectile thrown with initial velocity Vo makin an anle θ with the horizontal. Since the projectile describes motion in two dimensions therefore we can divide its velocity into two components i.e. horizontal and vertical component.

3 Horizontal Components: Accordin to the fiure the initial horizontal component can be written as, V ox = V o cos θ Since there is no horizontal unbalance force acts on the projectile durin motion therefore this component of velocity remains constant hence at any other instant final component will be, Vertical Component: V x = V ox = V o cos θ Initial velocity component of the velocity is iven by V oy = V o sinθ As there is a ravitational force responsible for the chane in the vertical component of velocity Voy therefore at any instant t its vertical component can be found as, By usin equation, V f = V i + at But V f = V y,v i = V oy = V o sinθ, and a = V y = V o sinθ t At the maximum heiht this vertical component becomes zero. 1) Maximum Heiht of Projectile The vertical distance of projectile motion at which the vertical component of velocity becomes zero is called maximum heiht H. By usin 3 rd equation of motion, 2aS = V f 2 V i 2

4 Here we have, V f = V y = 0; V i = V oy = V o sinθ; a = ; S = H; 2( )H = (0) 2 ( V o sinθ) 2 2H = V o 2 sin 2 θ H = V o 2 sin 2 θ 2 2) Time to reach Maximum Heiht Time to reach maximum heiht can be found by usin, V f = V i + at Here we have, t = V f V i a V f = V y = 0; V i = V oy = V o sinθ; a = ; We et, 3) Total time of Fliht t = 0 V osinθ t = V osinθ The time required for a body to reach the round is the same as the time to reach maximum heiht therefore, the total time of fliht can be written as,

5 T = 2V osinθ Also we can ive a mathematical prove of this total time with the help of 2 nd equation of motion, S = V i t t2 Here we have, S = H = 0; V i = V oy = V o sinθ; a = ; t = T; We et, 0 = (V o sinθ)t ( )T2 0 = (V o sinθ)t 1 2 T2 4) Rane of Projectile 1 2 T2 = (V o sinθ)t T = 2V osinθ Rane is the horizontal distance covered by the projectile in total time T usin equation for constant velocity, V = S t S = V t

6 Here we have, t = T = 2V osinθ ; V = V x = V ox = V o cosθ; S = R; We et, R = (V ocosθ)(2v o sinθ) R = V o 2 (2sinθcosθ) For Maximum Rane, R = V o 2 sin2θ The maximum horizontal distance covered by a projectile is called maximum rane. It depends on the value of sin2θ, sin2θ = 1 2θ = sin 1 (1) 2θ = 90 0 θ = 45 0 Therefore, for θ = 45 0 then we et maximum rane. R max = V o 2 sin2(45) R max = V o 2 sin90

7 R max = V o 2 5) Trajectory of Projectile The path on which the projectile moves is called trajectory of projectile. At any instant t the horizontal distance can be found as, S = V t Here we have, S = X, V = V o cosθ We et, X = V o cosθ t Or, t = X V o cosθ Similarly at the same instant t the vertical distance can be found by usin 2 nd equation of motion. S = V i t at2 Here we may write, S = Y; V i = V oy = V o sinθ; a = ; t = The above equation becomes, X V o cosθ ;

8 X Y = (V o sinθ) ( V o cosθ ) ( ) ( X V o cosθ ) Y = ( sinθ cosθ ) X ( 2V 2 o cos 2 θ ) X2 The equation can be written as, Y = (tanθ)x ( sec2 θ 2V2 ) X2 o Here it is clear that the terms in brackets are constant for particular projection and depends only upon initial values. θ, V o and are constant throuhout the motion and cannot be altered. So we are writin these constant as, tan θ = a; sec2 θ 2V o 2 = b; So we may amend the above equation as, Y = ax bx 2 There is an equation of a parabola therefore projectile describes a projectile motion.

9 Uniform Circular Motion If a body is movin with a constant speed in a circular path than the motion of a body is called uniform circular motion. Example: Motion of electron around the nucleus of the atom, motion of the satellite Anular Displacement: If a body is movin in a circular path than the anle covered by the body at the center of a circle is called anular displacement. Explanation: Let a body is movin on a circular path and at point A at an instant the line joinin this point to the center of the circle forms an anle θ 1 at the center with the horizontal line. Now after some time t the body reaches some other point B on the circle then the line joinin this point to the center of the circle will make an anle θ 2 ". Now the anular displacement covered by the body will be θ = θ 2 θ 1 Unit: the S.I unit of anular displacement is radians. Anular Velocity: Rate of chane of anular displacement is called anular velocity Unit: ω = θ t S.I. unit of anular velocity is radians per second; others are revolution per second or deree per second.

10 Direction: Anular velocity is a vector quantity and its direction can be found by riht hand rule if we curl the finers of our riht hand then thumb ives the direction of anular velocity. Anular Acceleration Rate of chane of anular velocity is called as anular acceleration. α = ω t If the velocity of the body is increasin then anular velocity and anular acceleration are in same direction and if velocity is decreasin then they are opposite. Tanential Velocity OR Linear Velocity: The velocity, which is obtained by dividin the arc lenth, covered by a body, by the time taken, is known as tanential velocity. This is the linear velocity of a body, whose direction can be found by drawin a tanent to the circle at any instant. Let a body is movin in a circle of radius r with anular velocity then at any point P its tanential velocity will be, Tanential Acceleration V = rω Rate of chane of tanential velocity is called tanential acceleration. OR If a body is not describin uniform circular motion then due to chane in the manitude of the velocity, acceleration is produced and is called tanential acceleration. Accordin to the definition of tanential velocity

11 V = rω V t = rω t a = rα Time Period: Time required to complete one vibration or rotation is called time period. T = 2π ω Centripetal Acceleration: The acceleration produced in a body due to continuous chanin in the direction of velocity is called centripetal acceleration. Explanation: If a body is movin in a circular path then the direction of velocity is chanin at every point of the circular path due to which acceleration produced in a body called centripetal acceleration. Its direction is perpendicular to the linear or tanential velocity and always direction towards the center of the circle. Mathematical Form: Consider a body of mass m movin in a circular path of radius r. Let at any instant t the body covers distance S from B to C. The velocity of a body at B is V1 and at C is V2. As the body is describin uniform circular motion then the manitude of its velocity is constant and let it is V. Therefore, the chane in velocity can be found as, V = V 2 V 1

12 If V1 and V2 are drawn from the same point P and a third vector is drawn from the arrowhead of V1 to the arrow head of V2 then this third vector will represent the velocity chane. The trianle ABC and the vector trianle PQR are similar hence we can write usin the property of similar trianles. BC AB = QR PQ S r = V V V = V S r For acceleration dividin above equation by t If t is small ( t 0) then But, V t = V S r t V lim t 0 t = V r lim S t 0 t lim t 0 V t = a lim t 0 S t = V Therefore, a = V(V) r

13 a c = V2 r DIRECTION: The direction of this acceleration would be the same as that of V. if t 0 then t 0 then the vectors V1 and V2 would be parallel and V would be perpendicular to the two velocity vectors and directed towards the center of the circle. Hence the direction of the centripetal acceleration will also be towards the center of the circle. CENTRIPETAL FORCE The force which keeps the body to move on a circular path is called centripetal force. EXAMPLES: Motion of satellite around the earth due to ravitational force, Motion of electron around the nucleus due to coulomb attractive force, MATHEMATICAL FORM: Consider an object of mass m is movin on a circular path of radius r then the centripetal force accordin to the Newton s second law will be F c = ma c But Hence, a c = V2 r F = mv2 r Also we may derive V = rω F = m(rω)2 r F = mrω 2 DIRECTION: Direction of the centripetal force is always towards the center of the circular path.