MOTION IN TWO OR THREE DIMENSIONS

Transcription

1 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

2 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 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

4 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

5 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

6 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

7 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

8 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

9 3.3 Projectile motion

10 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

11 3.3 Projectile motion: equations of motion (1) (2) v v at x x v t at 2 (3) v v 2 a( x x ) 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

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15 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

16 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

17 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

18 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

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