Any object that travels through a nonvacuous medium experiences a retarding force called drag. For small, aerodynamic objects that travel at high speeds, such as bullets, the drag force is negligible. For objects that are large, awkwardly shaped, nonuniform in material density, and/or travelling through highly viscous media, the drag force has a significant effect on the object's motion. This effect is dependent on several different factors that contribute to the overall motion, which can be combined into a value known as the drag coefficient. The drag coefficient is usually measured empirically, but it can be calculated from known quantities that characterise the problem at hand. The following steps will show how to derive the drag coefficient of an object of arbitrary shape in free fall.
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Draw a free-body diagram of the object's motion. Label the forces acting on the object. These will appear as vectors (arrows) that point in the direction of the force. In general, this will look like a drawing of your object with one vector pointed downward from the object's centre of mass, which represents the object's weight, and another vector pointed upward, which represents the resistive force acting on the object. The velocity vector in this case will be parallel to the downward force of the weight.
Write the relationship among the forces. Newton's laws of motion tell us that the sum of all of the forces acting on the object should equal zero. Therefore, our equation will look something like F = W + R = 0, where W is the object's weight, R is the resistive force (dependent on the drag coefficient), and F is the sum of all of the forces, which is equal to zero.
Substitute the weight and resistance relationships into the equation you wrote in Step 2. The weight relationship will be W = mg where m is the object's mass and g is the gravitational acceleration constant (g = 9.8m/s^2). The resistance relationship will be R = DpAv^2/2, where p is density of air (p = 1.29kg/m^3), A is the cross-sectional area of your object, v is the object's speed and D is the drag coefficient.
Solve the equation rewritten in Step 3 for the drag coefficient D, giving you the relationship D = (2mg)/(Pav^2).
Substitute the known values into the equation for the drag coefficient. The object's weight, cross-sectional area and terminal velocity should be given or empirically measured to complete this step. Perform the calculation to get a numeric value for the drag coefficient.
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