MOTION IN TWO OR THREE DIMENSIONS 3 Sections Covered 3.1 : Position & velocity vectors 3.2 : The acceleration vector 3.3 : Projectile motion 3.4 : Motion in a circle 3.5 : Relative velocity
3.1 Position & velocity vectors We can use vectors to indicate the position of a particle in space, in three dimensions r = xi Ƹ + yj Ƹ + zk
3.1 Position & velocity vectors Two vectors are needed to describe the motion of a particle through space Each vector represent the location of the points, P 1 and P 2 r 1 = x 1 i Ƹ + y 1 j Ƹ + z 1 k r 2 = x 2 i Ƹ + y 2 j Ƹ + z 2 k Their difference gives the displacement! r = r 2 r 1 As with 1-D motion we can define the velocities The average velocity v ave = r t = r 2 r 1 t 2 t 1 The instantaneous velocity r v = lim = r 2 r 1 t 0 t t 2 t 1 v = dr dt
3.1 Position & velocity vectors Components of velocity dr dx dy dz v i + j k dt dt dt dt The instantaneous velocity of a particle is always tangent to its path v v tan 2 2 x y v v y x v
3.1 Some definitions, summary Displacement Δr, or r A vector quantity denoting the direction and magnitude of a trajectory of an object Example: I went to Gainesville and back so my displacement is 0. Distance Δr, or r A scalar quantity: how far has the object traveled Example: I went to Gainesville and back so I traveled a longish distance of 2 r Average Velocity v The displacement over time v ave = r t Instantaneous velocity is just the instantaneous displacement over time v = dr Speed Magnitude of velocity can be used for instantaneous or average velocity s = r, dr t dt dt
3.2 The acceleration vector As with velocity we can use vectors to describe acceleration The average acceleration a ave = v t = v 2 v 1 t 2 t 1 The instantaneous acceleration v a = lim = dv t 0 t dt
3.2 The acceleration vector Any particle following a curved path is accelerating. It doesn t matter if its speed is constant Like velocity a expressed in components: a = dv = dv x i Ƹ + dv y j Ƹ + dv z k dt dt dt dt Can also be expressed in parallel and perpendicular components
3.3 Projectile Motion A body that when given an initial velocity follows a path influenced only by the force of gravity Simplifying assumptions we ll use Neglect air resistance Assume constant gravitational force Curvature of earth is flat
3.3 Projectile motion
3.3 Projectile motion Our strobe light example Red ball is just dropped no push Yellow ball is given a slight push to the right Motion can be analyzed in two dimensions separately (under simplifications used) In the horizontal direction (to the right = +x) The acceleration is zero a x = 0 Thus the velocity in that direction never changes! In the vertical direction is (up = +y) The acceleration is NOT zero a y = 9.8 m s 2 The velocity in that direction is constantly changing
http://en.wikipedia.org/wiki/kenny_powers_%28stuntman%29 3.3 Projectile motion: equations of motion (1) (2) v v at 0 0 0 1 2 x x v t at 2 (3) v v 2 a( x x ) 2 2 0 0 Along the vertical (y) direction 1) v y = v 0 sin α 0 gt 2) y = (v 0 sin α 0 )t 1 2 gt2 3) v 2 y = v 2 0y sin 2 α 0 2g(y f y 0 ) Time t Along the horizontal (x) direction 1) v x = v 0x cos α 0 2) x = (v 0 cos α 0 )t 3) v 2 x = v 2 0x cos 2 α 0
3.4 Motion in a circle We ll use vectors to analyze motion in a circle a= If the speed is constant then the trajectory is a perfect circle Note: that the velocity (vector) changes even though the speed doesn t A change in velocity implies an acceleration Regardless of whether that change is due to a change in magnitude or a change in direction dv dt
3.4 Motion in a circle at constant speed v s s v v R R a 1 ave v v1 t R s t v 1 In the limit where t 0, d v 1 v a = d2 r t dt 2 = v2 R ( r) if the speed is constant The radial acceleration s magnitude is a rad = v2 R. Its direction is always towards the center of the circle, it is given the name centripetal (center seeking) acceleration or more commonly radial acceleration
3.4 Motion in a non-uniform circle Non uniform circular motion If the speed (magnitude of velocity) varies around the circle then a component of the acceleration vector exits that is tangential to the velocity The centripetal component now changes around the circle
3.5 Relative Motion y A y B xa v B x B If a frame of reference is moving at a constant velocity then that frame is said to be an inertial frame of reference Netwon s Laws are valid only in inertial frames of reference