1. Physics 430: Lecture 4 Quadratic Air Resistance
Dale E. Gary
NJIT Physics Department

2. Linear Air Resistance Recap-1
When a projectile moves through the air (or other medium—such as gas or liquid), it experiences a drag force, which depends on velocity and acts in the direction opposite the motion (i.e. it always acts to slow the projectile).
Quite generally, we can write this force as , where the function f(v) can in general be any function of velocity.
At relatively slow speeds, it is often a good approximation to write
where flin and fquad stand for the linear and quadratic terms, respectively:
For linear air resistance, the equation of motion is or in terms of velocity, it is a first-order differential equation , which has component equations:
Equations of this form can be written:
where is the terminal velocity.
September 10, 2009

3. Linear Air Resistance Recap-2
Such equations are said to be in separable form (terms involving v on one side, and no dependence on v on the other side). Solutions of this particular form, e.g.
have exponential solutions:
which we then integrate to get x and y positions:
We can then combine these equations by eliminating t, to get a single equation for the trajectory:
Finally, we solved this for the range R, i.e. the value of x for which y = 0, valid for low air resistance:
Linear air resistance applies only to tiny projectiles or viscous fluids.
September 10, 2009

4. 2.4 Quadratic Air Resistance
For more normal size projectiles (baseball, cannon ball), it is the quadratic drag force that applies.
We are now going to follow exactly the same procedure, but starting with the quadratic form of the drag force:
The equation of motion (in terms of v) then becomes:
with component equations:
As we noted last time, these two equations are coupled, and are generally not solvable analytically (in terms of equations), although they can be solved numerically.
However, we can solve these equations for special cases of either solely horizontal motion (vy = 0), or solely vertical motion (vx = 0), in which case the equations become
Let’s look at these one at a time.
September 10, 2009

5. Horizontal Motion with Quadratic Drag-1
As before, we write the equation in separable form (move the terms involving v to one side). For the horizontal equation, it is trivial:
This equation is called a non-linear differential equation because one of the derivatives (the zeroth one, in this case) has a non-linear dependence. Such equations are significantly harder to solve, in general. In this case, however, the separable form allows us to integrate both sides directly
to get or , where I have introduced the
characteristic time, t, in terms of constants:
To find the position, we again integrate the velocity equation to get
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6. Horizontal Motion with Quadratic Drag-2
The final solutions for v(t) and x(t) are:
Graphs of these functions are:
They may look similar at first to the linear case, but now the velocity as
approaches zero much more slowly, like 1/t, so the position does not approach
some limiting value like in the linear case, but rather continues to increase
forever. If this sounds impossible, you are right. What really happens is that
as the speed drops, quadratic drag gets swamped by linear drag.
September 10, 2009

7. Vertical Motion with Quadratic Drag
We now consider motion solely in the vertical direction, governed by the equation of motion:
Before we write the vertical equation in separated form, however, we notice as before that the gravity force mg is balanced by the drag force cvy2 at terminal velocity
after which , i.e. the velocity becomes constant. In terms of vter, the separated form for the vertical equation is:
In this separated form, we can integrate both sides directly (assuming vo = 0).
Looking at the inside front cover of the book we find
which is what we have if we write x = v/vter. What the heck is arctanh?
September 10, 2009

8. Hyperbolic Functions—Problem 2.33(a)
Statement of the Problem:
The hyperbolic functions cosh z and sinh z are defined as follows:
for any z, real or complex. (a) Sketch the behavior of both functions over a suitable range of real values of z.
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9. Hyperbolic Functions—Problem 2.33(b)
Statement of the Problem, cont’d:
(b) Show that cosh z = cos(iz). What is the corresponding relation for sinh z?
Solution:
To do this part, you have to know the relations:
Then the solution is very easy: So
September 10, 2009

10. Hyperbolic Functions—Problem 2.33(c)
Statement of the Problem, cont’d:
(c) What are the derivatives of cosh z and sinh z? What about their integrals?
Solution:
The derivatives are:
The integrals are equally straightforward:
September 10, 2009

11. Hyperbolic Functions—Problem 2.33(d)
Statement of the Problem, cont’d:
(d) Show that cosh2 z – sinh2 z = 1.
Solution:
Since
and
the difference is
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12. Hyperbolic Functions—Problem 2.33(e)
Statement of the Problem, cont’d:
(e) Show that [Hint: One way to do this is to make the
substitution x = sinh z.]
Solution:
Making that substitution, we have dx = cosh z dz, so:
but
so we have shown that
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13. Vertical Motion with Quadratic Drag
Likewise, you can do Problem 2.34, which gives the definition:
and leads you through the steps needed to show
Now back to our equation:
The left side is
while the right side is just gt, so solving for v, we get
To get the position, integrate (see Prob. 2.34) to get
September 10, 2009

14. Example 2.5 A Baseball Dropped from a High Tower Solution
Find the terminal speed of a baseball (diameter D = 7 cm , mass m = 0.15 kg). Make plots of its velocity and position for the first six seconds after it is dropped from a tall tower.
Solution
Recall that the constant c can be written c = gD2., where g = 0.25 Ns2/m2. So
which is nearly 80 mph.
You can sketch the velocity and position, or you can calculate it in Matlab. Here are the plots. As expected, the velocity increases more slowly than it would in a vacuum under gravity (dashed line), and approaches vter = 35 m/s (dotted line). As a consequence, the position is less than the parabolic dependence in vacuum.
September 10, 2009

15. Quadratic Drag with Horizontal and Vertical Motion
As we said before, the general problem of combined horizontal and vertical motion yields a set of coupled equations
where now we take y positive upward.
The projectile does not follow the same x and y equations we just derived, because the drag in the x direction slows the projectile and changes the drag in the y direction, and vice versa. In fact, these equations cannot be solved analytically at all! The best we can do is a numerical solution, but that requires specifying initial conditions. That means we cannot find the general solution—we have to solve them numerically on a case-by-case basis.
Let’s take a look at one such numerical solution.
September 10, 2009

16. Example 2.6 Trajectory of a baseball Solution
The baseball of the previous example is thrown with velocity 30 m/s (about 70 mph) at 50o above the horizontal from a high cliff. Find its trajectory for the first 8 s of flight and compare with the trajectory in a vacuum. If the same baseball were thrown on the same trajectory on horizontal ground, how far would it travel (i.e. what is its horizontal range)?
Solution
First, what are the initial conditions for the position and velocity? For the position, we are free to choose our coordinate system, so we certainly would choose xo = 0 and yo = 0 at t = 0. For the velocity, the statement of the problem gives the initial conditions vxo = vocos q = 19.3 m/s, vyo = vosin q = 23.0 m/s. Using these values, we need to write a program in Matlab that performs a numerical solution to the equations
for the time range 0 < t < 8 s. We will use the routine ode45 (ode stands for ‘ordinary differential equation’).
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17. Example 2.6, cont’d Solution, cont’d
First we have to write a function that will be called by ode45. The heart of that routine is quite simple, just write expressions for the two equations:
= [block 2]
Here, v is the velocity vector, so v(1) is the horizontal velocity vx and v(2) is the vertical velocity vy. Before these equations will work, we have to define the constants, g, c, and m. Recall that c = gD2.
= [block 1]
The last step is to name the function and indicate the inputs and outputs. ODE45 specifies that the function must have two inputs—the limits of the independent variable (time in this case), and the array of initial conditions (start velocity in x and y in this case).
After saving this function as quad_drag.m, we call ODE45 with
Vdot_x = -(c/m)*sqrt(v(1)^2+v(2)^2)*v(1);
Vdot_y = -g-(c/m)*sqrt(v(1)^2+v(2)^2)*v(2);
m = 0.15; % Mass of baseball, in kg
g = 9.8; % Acceleration of gravity, in m/s
diam = 0.07; % Diameter of baseball, in m
gamma = 0.25; % Coefficient of drag in air at STP, in Ns^2/m^2
c = gamma*diam^2;
function vdot = quad_drag(t,v) …
[block 1]
[block 2]
vdot = [vdot_x; vdot_y];
[T,V] = ode45('quad_drag',[0 8],[19.3; 23.0]);
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18. Example 2.6, cont’d Solution, cont’d
The arrays T and V that are returned are the times and x and y velocities, but what we need is the trajectory, i.e. the x and y positions. For those, we have to integrate the velocities. There is probably an elegant way to do this in Matlab, but I wrote a simple (and rather inaccurate) routine to do that, given the T and V arrays:
Save this as int_yp.m, and then call it by
which returns the position array [pos(1,:) is x, pos(2,:) is y].
All that remains is to plot the trajectory ( i.e. pos(1,:) vs. pos(2,:) ).
function y = int_yp(t,yp)
n = length(t);
y = yp;
y(1,:) = [0 0];
for i=1:n-1
dt = t(i+1)-t(i);
dy = yp(i,:)*dt;
y(i+1,:) = y(i,:)+dy;
end
y = y(1:n-1,:);
pos = int_yp(T,V);
plot(19.3*T,23.0*T-4.9*T.^2); % Plot vacuum case
hold on
plot(pos(:,1),pos(:,2),'color','red'); % Overplot quadratic drag case
hold off
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19. Example 2.6, cont’d Solution, cont’d
Here is the resulting plot (somewhat improved by labels).
Note that the range is about
60 m, much shorter than the
equivalent trajectory in a
vacuum.
Note also that the baseball
does not reach quite as high
as in a vacuum, and reaches
its peak earlier.
You will be given homework
problems in which I will ask
you to try your hand at such
numerical solutions and plotting.
I will help you learn these very
useful skills, or you can make
use of the Matlab helpers.
September 10, 2009

20. 2.5 Motion of a Charge in a Uniform Magnetic Field
You may recall from Physics 121 that the force on a charge moving in a magnetic field is
where q is the charge and B is the magnetic field strength. The equation of motion then becomes
which is a first-order differential equation in v.
In this type of problem, we are often free to choose our coordinate system so that the magnetic field is along one axis, say the z-axis:
and the velocity can in general have any direction Hence,
and the three components of the equation of motion are:
September 10, 2009

21. Motion of a Charge in a Uniform Magnetic Field-2
This last equation simply says that the component of velocity along B,
vz = const. Let’s now focus on the other two components, and ignore the motion along B. We can then consider the velocity as a two-dimensional vector (vx, vy) = transverse velocity.
To simplify, we define the parameter w = qB/m:, so the equations of motion become:
We will take the opportunity provided by these two coupled equations to introduce a solution based on complex numbers.
As you should know, a complex number is a number like z = x + iy, where i is the square root of -1. Let us define:
and then plot the value of h as a vector in the
complex plane whose components are vx and vy.
imaginary
part
real part vx vy
h = vx+ ivy
h = vx + ivy
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22. Motion of a Charge in a Uniform Magnetic Field-3
Next, we take the time derivative of h: or
So the equation in terms of this new relation has the same form we saw in the previous lecture for linear air resistance, with the familiar solution
The only difference is that this time the argument of the exponential is imaginary, but it turns out that this makes a huge difference.
Before we can discuss the solution in detail, however, we need to introduce some properties of complex exponentials, which we will do next time.
September 10, 2009