2023_Lecture11_12_13

PHY 1321/PHY1331 Principles of Physics I Fall 2023 Dr. Andrzej Czajkowski 111 Concept of a Particle (Material Point) We can effectively describe the TRANSLATIONAL MOTION of a body using the idealization of a particle (dimensionless point of the same mass as a body (material point)) By doing this, we are neglecting various effects, which are related to the body’s physical dimensions (such as air-resistance, buoyancy, and other possible body rotational effects). For many of the problems discussed in our introductory course, this is an excellent approximation and we can describe many aspects of motion using this simplified model. Fundamental definitions for 1-D motion: Position Displacement Fundamental definitions for 3-D motion (vectors!): Position: Displacement: Average velocity Instantaneous velocity definition Average acceleration definition Instantaneous acceleration definition M Particle (material point) of mass M 2 2 0 0 lim lim ) ( dt r d dt dr dt d dt dv t v a t a t avg t = ÷ ÷ ø ö ç ç è æ = = D D = ® D ® D = • ® D ® D = = D D = = r dt dr t r v t v t avg t lim lim 0 0 ) ( t x erval int time nt displaceme v v avg D D = = ñ á = 1 2 1 2 t t v v t v erval int time velocity in change a a avg - - = D D = = ñ á = ^ ^ ^ ^ ^ ^ ^ ^ ^ z y x z y x i z i y i x k z j y i x k r j r i r r + + = + + = + + = ) , , ( ) , , ( z y x r r r r z y x = = ^ ^ ^ ^ 1 2 ^ 1 2 ^ 1 2 ^ 1 ^ 1 ^ 1 ^ 2 ^ 2 ^ 2 1 2 ) ) ( ) ( ) ( ) ( ) ( k z j y i x k z z j y y i x x r k z j y i x k z j y i x r r r r D + D + D = - + - + - = D + + - + + = D - = D 2 2 0 0 lim lim ) ( dt x d dt dx dt d dt dv t v a t a t avg t = ÷ ø ö ç è æ = = D D = ® D ® D = • ® D ® D = = D D = = x dt dx t x v t v t avg t lim lim 0 0 ) ( t x erval int time nt displaceme v v avg D D = = ñ á = t s D = = interval time distance total speed average 1 2 1 2 t t v v t v erval int time velocity in change a a avg - - = D D = = ñ á = i f x x x - = D

PHY 1321/PHY1331 Principles of Physics I Fall 2023 Dr. Andrzej Czajkowski 114 Suggested Problems 1. Particle moves from initial position r i to final position r f in 2.0s a) Find average velocity vector during this move, b) Find the magnitude of the average velocity vector 2. Time dependent vector r(t) describes the position of the Particle A at any instant of time: a) Find x ,y, z components of instantaneous velocity v(t) vector. b) Find magnitude of the velocity vector at time t=2s 3. An ion’s position vector is initially r = 5.0 i - 6.0 j +2.0 k , and 10s later it is r = -2.0 i + 8.0 j -2.0 k , all in meters. What is its average velocity during this time interval ? 4. The position of an electron is given by r = (3.00t) i – (4.00t) j +(2.00) k, / t in second and r in meters / a) What is the electron’s velocity v ? b) What is electron’s velocity at t=2s (in unit-vector notation)? c) What is a magnitude and angle relative to the positive direction of the x axis? 5 A radar station detects an airplane approaching directly from the east. At first observation the range to the plane is 360m at 40 o above the horizon. The airplane is tracked for another 123o in the vertical east-west plane, the range at final contact being 790 m. Find the displacement of the airplane during the period of observation. 6 An object A moves along the x-axis according to the equation: x(t) = 3.0t 2 +3.3t+10.4, where x is in meters, and t is in seconds. The other object (object B) moves along the x-axis with constant speed +9 m/s. When does A have the same instantaneous velocity as B ? State your answer to nearest 0.01s. 7 Particle A moves along the line y=30 m with a constant velocity v of magnitude 3.0m/s and directed parallel to the positive x axis. Particle B starts at origin with zero speed and constant acceleration a (of magnitude0.40m/s2) at the same instant that particle A passes the y axis. What angle θ between a and the positive axis would result in a collision between these two particles? 8 The ball is shot vertically with the initial velocity of 7 m/s from the 13m level. What is the maximum height it will reach? Neglect the air resistance, and take g=9.8 m/s 2 . Give the answer to the nearest tenth of a meter. 9 The ball is shot vertically with the initial velocity of 3 m/s from the 10m level. How long it will take for the ball to reach the ground? Neglect the air resistance, and take g=9.8 m/s 2 . Give the answer to the nearest hundredth of a second. ^ ^ ^ 2 2 6 2 ) ( k t j t i t t r i + - = ^ ^ ^ ^ ^ ^ 7 4 ; 6 2 k j i r k j i r f i - + = + - =

PHY 1321/PHY1331 Principles of Physics I Fall 2023 Dr. Andrzej Czajkowski 117 Two-dimensional motion: projectile motion In projectile motion, the horizontal motion and the vertical motion are independent of each other: They do not affect each other and may be considered separately THE TRAJECTORY: The path of a projectile that is launched at t=0 (x 0 =0, y 0 =0) with initial velocity v 0 (v 0x ,v 0y ) Initial velocity and velocities at various points along its path are shown. R (Range) is the horizontal distance the projectile has traveled when it returns to its launch height

PHY 1321/PHY1331 Principles of Physics I Fall 2023 Dr. Andrzej Czajkowski 120 Effects of the Air: Real Projectiles VACUUM AIR RANGE 177 m 99 m MAX H 77 m 53 m T.O.F. 7.9 s 6.6 s In reality, projectile motion is more complex than we assumed. Depending on the shape of the projected object, there is a certain amount of air resistance proportional to the square of the speed or to the speed) Real ballistic trajectories are not parabolic! V=600 m/s theta=60 Uniform Circular Motion A Particle is in a uniform circular motion if it travels around a circle in a circular arc at a constant (uniform) speed Even though the speed is constant, the particle is accelerating. Magnitude of centripetal acceleration Period of rotation (T) r v a a 2 = = v r T p 2 =

PHY 1321/PHY1331 Principles of Physics I Fall 2023 Dr. Andrzej Czajkowski 123 At some instant both A and B measure the P position and velocity. How will their observation relate to each other? Taking derivative of both sides of this equation we get: Acceleration in the Relative Motion What about the acceleration? To find out let’s take derivative of the equation describing velocity relationship: CONCLUSION: Observers in two different reference frames, moving at constant velocity relative to each other, will measure the same acceleration for moving particle. Relative motion in 3-D: We may easily generalize these 1-D results to 3-D PB BA PA x x x + = dt dx dt dx dt dx BA PB PA + = BA PB PA v v v + = PB PA PB PA const v BA PB PA a a dt dv dt dv dt dv dt dv dt dv BA = Û = Û + = = BA PB PA v v v + = PB BA PA r r r + = BA PB PA v v v + = PB PA PB PA const v BA PB PA a a dt v d dt v d dt v d dt v d dt v d BA = Û = Û + = =