y 1 y 2 x 1 x 2 Using Cartesian Coordinates to Describe 2D Motion motion path described by the particle on 2D X Y P 1 P 2 i = (1,0) j = (0,1) R = x i + y j = (x, 0) + (0, y) = (x, y) R(t) = x(t) i + y(t) j = (x(t), 0) + (0, y(t)) = (x(t), y(t))... v(t) = R(t) = (x(t), y(t))  always in the tangent direction of the motion path..... a(t) = v(t) = (x(t), y(t)) R1R1 R2R2 v 1 v 2

motion path described by the particle on 2D X Y P1P1 R1R1 Using Polar Coordinates to Describe 2D Motion R(t) = r(t) e r (t) r1r1... v(t) = R (t) = r(t) e r (t) + r(t) e r (t) = v r (t) e r (t) + v  (t) e  (t)  always in the tangent direction of the motion path a(t) = v (t) = r(t) e r (t) + 2r(t) e r (t) + r(t) e r (t) = a r (t) e r (t) + a  (t) e  (t)  X erer ee  information is already contained in e r, so e  does not show up in R(t)  If R 1 (t) = r 1 (t) e r, R 2 (t) = r 2 (t) e r ’, R 1 (t) + R 2 (t)  (r 1 (t) + r 2 (t)) e r since e r (t)  e r ’ (t). So, how do we do vector algebra in polar coordinates? er’er’ e’e’ R2R2 angle accelerationradial acceleration v 1 v 2

motion path described by the particle on 2D X Y P1P1 Using Tangential/Normal Coordinates to Describe 2D Motion It is no longer required for a fixed origin. P1P1 Using T/N coordinate, one is only interested in the velocity and acceleration of the moving particles. Hence, there is no position vector. It is best to imagine you are traveling along the motion path (see figure). T/N coordinate system relies on the arc-length variable, S, to measure the velocity and acceleration. Given any instant, say t 0, for any t > t 0 S(t) measures the distance traveled by the particle From t 0 to t. To define the speed, consider t > t 0, the limit S(t) = lim t  t 0 S(t)/(t – t 0 ) defines the velocity at point P 1. Imagine the limit as if there are t 3 > t 2 > t 1 > … > t 0 and measure the corresponding S(t 3 ), S(t 2 ), S(t 1 ), … In the limit t  t 0, S(t)  0 and t – t 0  0. But its ratio measures the speed of the velocity. S(t 3 ) S(t 2 ) S(t 1 )

P1P1 Using Tangential/Normal Coordinates to Describe 2D Motion But, S(t) measures the speed, which is the magnitude of the velocity. The direction of the velocity has not yet been specified. So, where is it? Recall from Cartesian, Polar coordinate systems, the velocity is always in the tangent direction of the motion path. Therefore, the velocity expressed in T/N coordinate is v(t) = S(t) e t (t) magnitude of velocity The acceleration follows immediately by differentiation: a(t) = v(t) = S(t) e t (t) + S(t) e t (t) = a t (t) e t (t) + a n (t) e n (t) tangent acceleration measures how fast S(t) changes in time. (tangential) direction of the velocity (see the red arrow) etet normal acceleration, which describes the change in the direction of motion and points toward the center of curvature at the given position of the motion path (see the blue arrow). enen

motion path described by the particle on 2D X Y P1P1 P2P2 Using Tangential/Normal Coordinates to Describe 2D Motion etet enen etet enen