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Unformatted text preview: 1 The Effect of Air Resistance Harvey Gould Physics 127 2/06/07 I. INTRODUCTION I simulated the motion of a falling body near the earth’s surface in the presence of gravity and air resistance. I used the Euler, Euler-Cromer, and Euler-Richardson algorithms and considered both a linear and quadratic dependence of the drag force on the velocity. Of particular interest is the rise and fall of a pebble and the relative times of ascent and descent. II. METHOD As shown in the text, Newton’s laws of motion can be written as two coupled first-order differential equations. The Euler algorithm for the numerical solution of the latter equations can be written in the form v n +1 = v n + a n Δ t (1a) y n +1 = y n + v n Δ t. (1b) The Euler-Cromer algorithm modifies Eq. (1) slightly and can be expressed as v n +1 = v n + a n Δ t (2a) y n +1 = y n + v n +1 Δ t. (2b) A more sophisticated version of the Euler algorithm evaluates the acceleration and the velocity at the time Δ t/ 2 = t + Δ t/ 2. The Euler-Richardson algorithm can be written as follows: a m = F ( y m , v m , Δ t/ 2) /m (3a) v m = v n + a n Δ t/ 2 (3b) y m = y n + v n Δ t/ 2 , (3c) 2 and v n +1 = v n + a m Δ t (4a) y n +1 = y n + v m Δ t. (4b) The total force on a falling body in the presence of air resistance can be written in the convenient form F =- mg (1- v 2 /v 2 t ) (quadratic drag force) (5a) or F =- mg (1- v/v t ) (linear drag force) (5b) The signs in Eq. (5) are for a falling body with the vertical coordinate increasing upward.The signs in Eq....
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This note was uploaded on 10/05/2011 for the course ME me352 taught by Professor Koraykadirsafak during the Spring '11 term at Yeditepe Üniversitesi.
- Spring '11