Falling with Air Resistance- Terminal Velocity http://www.physicsclassroom.com/mmedia/newtlaws/efff.cfm Terminal Velocity
Resistive (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 medium – the size and shape of the object – the speed of the object When equilibrium is reached between the drag force and other forces acting on the object, then the object travels with a terminal velocity.
Magnitude of the Resistive Force, R R can depend on v in a variety of ways. Here are two: a)Slow motions or small objects R = b v b)Large objects R = bv + cv 2 Note: b and c are constants, that depend on the medium and the size and shape of the object.
Example 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
For the same example find the terminal velocity, v t. a= dv = g- b v dt m When t=0 then v=0, so a= g When a=0, terminal velocity is reached: a= 0= dv = g- b v v t = mg dt m b
For the same example the general expression for v is: v= v t (1-e -t/ ) The time constant = m/b, is the time by which the sphere reaches 63.2% of its terminal speed.
R = ½ D Av 2 – D : drag coefficient (for a sphere D=0.5) – : density of air – A : cross-sectional area of the object measured in a plane perpendicular to v (for a sphere A= r 2 ) – v : speed of the object Air Drag at High Speeds
Example 2: Find the expression for acceleration for a falling object with air resistance. Ans: a= g-(D A) v 2 2m
For the same example find the terminal speed. Ans: