[Physics] Projectile Motion with Air Resistance and Wind

classical-mechanicscomputerdragprojectile

I am wondering how the general kinematics equations would change in the following situation.

If an object were fired out of a cannon, or some sort of launcher, so that it had both an initial velocity and an initial angle, and air resistance is taken into account, what would be the equations for the x and y components of the position, velocity, and acceleration.

Furthermore, I am wondering how these equations would change if there were also a wind blowing at an angle. In essence, what I would like to know is how to rewrite the kinematics equations to take into account the air resistance and moving wind and the terminal velocity.

The reason I want to know this is that I am writing a program to model this behavior, but I first need to know these equations.

Also, if possible, could someone provide some help on finding equations for the maximum height the projectile reaches, as well as the distance it travels before it hits the ground? I would like both of these to be values the user of the program can find if desired.

Oh, and in the scenario of the wind, it can blow from any angle, which means it will affect the x and y velocities and either augment them or lessen them depending on the angle at which it blows. So I guess another request is an explanation of how to obtain the set of equations (position, velocity, acceleration) for the x direction based on whether the wind angle is helpful or hurtful, and how to obtain the set of equations for the y direction, based again on whether the wind is helpful or hurtful.

I would naturally have a constraint on the wind velocity so that the object would always inevitably hit the ground, so the force of the wind in the y-direction, if it were blowing upwards, would have to be less than the force of gravity of the object, so that it still fell. Sorry, I know I'm asking a lot, it's just that I really want to understand the principles behind this. Any help at all here would be very much appreciated, but if possible, could whoever responds please try to address all of my questions, numerous though they are?

Oh, one final note. As this is being written in a computer program (python 2.7.3, to be exact), I cannot perform any integration or differentiation of the functions. Instead, I will need to create a small time step, dt, and plot the points at each time step over a certain interval. The values of the radius of the object, its mass, its initial velocity and angle, the wind velocity and angle, and dt can all be entered by the user, and the values of wind angle and wind velocity are defaulted to 0, the angle is defaulted to 45 degrees, and dt is defaulted to 0.001, although these values can be changed by the user whenever they desire.

Thanks in advance for any help provided!

Best Answer

As mentioned in the comments, this is an extremely complex problem if you intend to consider every possible aspect. However, for a general estimation, you can use the relatively simple methods described in this document to begin calculating the effects of air drag on projectiles.

Note that in the document cited, they make the assumption that the air is not moving, and begun their derivation from $f = Dv^2$, and this $v$ was relative to the air and therefore the following equations simply used the velocity of the ball. For the more complex case where the air is moving as well, you will need to account for this change and make sure that the x and y components of the force due to drag are calculated using the relative velocity of the projectile through the now-moving air.

Also worth noting is the fact that if the wind direction changes, the effective footprint of your projectile will change, thus changing $D$ and therefore the force due to drag. If you are willing to make a reasonable approximation for the average footprint of your projectile, however, this will likely yield a result that is accurate enough for your purposes.

Hope this helps!

Related Solutions

[Physics] Projectile Motion with Drag

In two dimensions, Newton's second law can be written in vector form as $$ \mathbf F_\mathrm{net} = m\mathbf a $$ In this case, the net force is $$ \mathbf F_\mathrm{net} = m\mathbf g - kv^2\frac{\mathbf v}{v} = m\mathbf g - kv\mathbf v $$ so the equation of motion is $$ m\mathbf a = m\mathbf g - kv\mathbf v $$ In components, if we choose the positive $y$ direction to be vertical, and using $v = \sqrt{v_x^2 + v_y^2}$ as you point out, we obtain $$ ma_x = -k\sqrt{v_x^2+v_y^2} v_x, \qquad ma_y = -mg-k\sqrt{v_x^2+v_y^2}v_y $$ as you can see, these differential equations are coupled; the $x$ equation involves $v_y$ and the $y$-equation involves $v_x$ unlike the case in which there is no drag. You should be able to numerically solve these simultaneous equations pretty easily on Mathematica.

In particular, you can solve these equations by specifying the initial position $\mathbf x(0) = (x(0), y(0)$ and the initial velocity $\mathbf v(0) = (v_x(0), v_y(0)) = (v(0)\cos\theta, v(0)\sin\theta)$ where $\theta$ is the initial angle at which the projectile is launched.

Projectile Motion – How to Add Air Resistance to Projectile Motion

With drag force $- \alpha \left|{\dot{\bf r}}\right| {\dot{\bf r}}$ and gravitational force $-mg {\bf {\hat{y}}}$, the equations of motion are (see my answer to this question) $$ \begin{align} {\ddot{x}} &= - \beta {\dot{x}} \sqrt{{\dot x}^2+{\dot y}^2} \\ {\ddot{y}} &= - g - \beta {\dot{y}} \sqrt{{\dot x}^2+{\dot y}^2} \end{align} $$ where $\beta = \alpha / m$, $\alpha = \rho C_d A / 2$, $\rho$ is the air density, $C_d$ is the drag coefficient, and $A$ is the cross-sectional area of the projectile.

I've never seen the above equations solved analytically. Here's some Python code that plots the solution with and without air resistance. To run, copy into a text file myFile.py and run

python myFile.py

from the command line.

from pylab import *
import math

# Physical constants
g = 9.8
m = 1.0
rho = 1.0
Cd = 1.0
A = math.pi * pow(0.01,2.0)
alpha = rho * Cd * A / 2.0
beta = alpha / m

# Initial conditions
X0 = 1.0
Y0 = 0.0
Vx0 = 70.0
Vy0 = 80.0

# Time steps
steps = 1000
t_HIT = 2.0*Vy0/g
dt = t_HIT / steps

# No drag
X_ND = list()
Y_ND = list()

for i in xrange(steps+1):
  X_ND.append(X0 + Vx0 * dt * i)
  Y_ND.append(Y0 + Vy0 * dt * i - 0.5 * g * pow(dt * i,2.0))

# With drag
X_WD = list()
Y_WD = list()
Vx_WD = list()
Vy_WD = list()

for i in xrange(steps+1):
  X_ND.append(X0 + Vx0 * dt * i)
  Y_ND.append(Y0 + Vy0 * dt * i - 0.5 * g * pow(dt * i,2.0))

# With drag
X_WD = list()
Y_WD = list()
Vx_WD = list()
Vy_WD = list()

X_WD.append(X0)
Y_WD.append(Y0)
Vx_WD.append(Vx0)
Vy_WD.append(Vy0)

stop = 0
for i in range(1,steps+1):
  if stop != 1:
    speed = pow(pow(Vx_WD[i-1],2.0)+pow(Vy_WD[i-1],2.0),0.5)

    # First calculate velocity
    Vx_WD.append(Vx_WD[i-1] * (1.0 - beta * speed * dt))
    Vy_WD.append(Vy_WD[i-1] + ( - g - beta * Vy_WD[i-1] * speed) * dt)

    # Now calculate position
    X_WD.append(X_WD[i-1] + Vx_WD[i-1] * dt)
    Y_WD.append(Y_WD[i-1] + Vy_WD[i-1] * dt)

    # Stop if hits ground
    if Y_WD[i] <= 0.0:
      stop = 1

# Plot results
plot(X_ND, Y_ND)
plot(X_WD, Y_WD)
show()

Best Answer

Related Solutions

[Physics] Projectile Motion with Drag

Projectile Motion – How to Add Air Resistance to Projectile Motion

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