3Resistive (Drag) Forces When an object moves through a medium (liquid or gas), the medium exerts a resistive force, R, on the object, opposing its direction of motion.The magnitude of the drag force depends on:the mediumthe size and shape of the objectthe speed of the objectWhen equilibrium is reached between the drag force and other forces acting on the object, then the object travels with a terminal velocity.
4Magnitude of the Resistive Force, R R can depend on v in a variety of ways.Here are two:a)Slow motions or small objectsR = b vb)Large objectsR = bv + cv2Note: b and c are constants, that depend on the medium and the size and shape of the object.
5Example 1: Slow speed, small object Find the expression for acceleration in terms of v, b, g, m.Answer: a= dv = g- b v dt m
6For the same example find the terminal velocity, vt. a= dv = g- b v dt mWhen t=0 then v=0, so a= gWhen a=0, terminal velocity is reached:a= 0= dv = g- b v vt= mg dt m b
7For the same example the general expression for v is: v= vt (1-e-t/t )The time constantt = m/b, is the time by which the sphere reaches 63.2% of its terminal speed.
8R = ½ DrAv2 Air Drag at High Speeds D : drag coefficient (for a sphere D=0.5)r : density of airA : cross-sectional area of the object measured in a plane perpendicular to v(for a sphere A= pr2 )v : speed of the object
9Example 2: Find the expression for acceleration for a falling object with air resistance. Ans: a= g-(D rA) v22m
10For the same example find the terminal speed. Ans: