Aerodynamic Drag Forces

The only forces we assume are from gravity. The exercise will ignore the change of gravitational acceleration for motion near the surface of the earth. The forces exerted on a mass, m, are W , the weight and the drag force D .

F = D + W

a=D+W/ m

In equation 1a the total force is, F . We want to determine the acceleration, a. Knowing the acceleration we follow the trajectory of m in steps of time, dt. This must be done numerically. Consider the displacement, r(t), and velocity, v(t), at any given moment of time, t,

dv = adt (2a).

v(t + dt) = v(t) + dv (2b)

dr = v(t)dt (2c)

r(t + dt) = r(t) + dr (2d)

You can employ the steps in equations 2a-2d in a computer code you write or use one of the mathematical packages available.

By selecting a step size, dt you are deriving an analytical solution(by quadrature) according You keep track of the total time.

Following a 45 caliber bullet. Use MKS units!

(A) A 45 caliber bullet has a mass of 13g and a cross sectional area of A = 1.06 104m2. Assume the muzzle velocity is v = 275m/s. For this cal- culation ignore drag. What is the maximum height, in meters, and the time of flight to maximum height in seconds? At what time after firing does the bullet return to ground? Assume the initial velocity is vertical.

(B) The drag equation is

D = C_d(v)pv²A / 2

(In equation 3) the coefficent of drag for this type of bullet depnds on the speed, v, C_d(v) and on air density, is the air density.

The direction of the drag force is opposite the velocity of the bullet. Note that if = 0 we are back in the introductory physics situation.

When you solve the problem, using whatever algorithm you chose, check, as a matter of principle, that if = 0 your algorithm gives you the answer in (A).

In order to apply equation 3 you should consider the importance of the dependence of the air density, (h)

(Question1) What is the maximum height the bullet reaches and the time to maximum height if it starts from sea level?

(Question2) What is the time it takes for the bullet to fall back to the ground and its speed?

(Question3) Suppose the bullet is fired from Colorado Springs(altitude 1840m). What is the maximum height of the bullet and the time to maximum height?

(Question4) What is the time it takes for the bullet to fall back to the ground and its speed in Colorado Springs?

(Question5) What is the terminal velocity of the bullet?