Physics 430: Lecture 4 Quadratic Air Resistance

Dale E. Gary

NJIT Physics Department

September 10, 2009

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.

Linear Air Resistance Recap-1

vf ˆ)(vf

quadlin2)( ffcvbvvf

2quadlin and cvfbvf

vgr bmm vgv bmm

yy bvmgvm xx bvvm

m

b

vv

v

m

b

v

v

y

y

x

x

ter

;

b

mgv ter

September 10, 2009

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.

Linear Air Resistance Recap-2

dtm

b

vv

dvdt

m

b

v

dv

y

y

x

x

ter

;

/o

txx evv )1( /

ter/

o tt

yy evevv

/1)( textx )1()( /

teroter t

y evvtvty

bm /

oter

o

tero 1lnxx

y

v

xvx

v

vvy

ter

ovac 3

41

v

vRR y

September 10, 2009

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.

2.4 Quadratic Air Resistance

2quad cvf

vgv ˆ2cvmm

yyxy vvvcmgcvmgvm 222 xyxx vvvccvvm 222

motion] l[horizonta 2cvvm motion] [vertical 2cvmgvm

September 10, 2009

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, , in terms of constants: .

To find the position, we again integrate the velocity equation to get

Horizontal Motion with Quadratic Drag-1

dtm

c

v

dv

2

tv

vtd

m

c

v

vd02

o

m

ct

vv

11

o /1)( o

t

vtv

ocv

m

0] [if )/1ln(/1

)( oo0

0o

xtvtd

t

vxtx

t

September 10, 2009

The final solutions for v(t) and x(t) are:

Graphs of these functions are:

drag] quadratic[for / ocvm

Horizontal Motion with Quadratic Drag-2

ovxv

t

t

x

/1)( o

t

vtv

)/1ln()( o tvtx

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 approachsome limiting value like in the linear case, but rather continues to increaseforever. If this sounds impossible, you are right. What really happens is thatas the speed drops, quadratic drag gets swamped by linear drag.

t

x

September 10, 2009

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 cvy

2 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 findwhich is what we have if we write x = v/vter. What the heck is arctanh?

Vertical Motion with Quadratic Drag

c

mgv ter

0yv

dtgvv

dv

2ter

2 /1

motion] [vertical 2cvmgvm

tv

tdgvv

vd00 2

ter2 /1

xx

dxarctanh

1 2

September 10, 2009

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.

Hyperbolic Functions—Problem 2.33(a)

2sinh and

2cosh

zzzz eez

eez

21

zcosh

z

1 2

ze

2

ze

21

zsinh

z21

2

ze

2

ze

September 10, 2009

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

Hyperbolic Functions—Problem 2.33(b)

ziee

ii

ee

i

eeiz

zzzziziizi

sinh222

sin)()(

zeeee

izzziziizi

cosh22

cos)()(

i

eex

eex

ixixixix

2 sin and

2cos

izii

izz sin

sinsinh

September 10, 2009

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:

Hyperbolic Functions—Problem 2.33(c)

zeeeedz

d

dz

zd zzzz cosh)()(sinh

21

21

zeeeedz

d

dz

zd zzzz sinh)()(cosh

21

21

zdzz sinhcosh zdzz coshsinh

September 10, 2009

Statement of the Problem, cont’d: (d) Show that cosh2 z – sinh2 z = 1.

Solution: Since

and

the difference is

Hyperbolic Functions—Problem 2.33(d)

)2()2(2

cosh 224122

41

2

2 zzzzzzzz

eeeeeeee

z

1sinhcosh 21

2122 zz

)2()2(2

sinh 224122

41

2

2 zzzzzzzz

eeeeeeee

z

September 10, 2009

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

Hyperbolic Functions—Problem 2.33(e)

xx

dxarcsinh

1 2

xzxz arcsinh sinh

zdzzzz

dzz

z

dzz2222 sinhsinhcosh

cosh

sinh1

cosh

xx

dxarcsinh

1 2

September 10, 2009

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

Vertical Motion with Quadratic Drag

z

zz

cosh

sinhtanh

tv

tdgvv

vd00 2

ter2 /1

xx

dxarctanh

1 2

terter

/

0ter

/

0 2ter0 2ter

2

arctanh arctanh

1/1

ter

ter

v

vvxv

x

dxv

vv

vd

vv

vvv

terter tanh)(

v

gtvtv

ter

2ter coshln)(

v

gt

g

vty

September 10, 2009

Example 2.5 A Baseball Dropped from a High Tower

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 = D2., where = 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.

2

ter 2 2 2

(0.15 kg)(9.8 m/s )35 m/s

(0.25 Nm /s )(0.07 m)

mgv

c

September 10, 2009

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.

yyxy vvvcmgvm 22 xyxx vvvcvm 22

yyxy vvvcmgvm 22 xyxx vvvcvm 22

September 10, 2009

Example 2.6 Trajectory of a baseball

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 = 19.3 m/s, vyo = vosin = 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’).

yyxy vvvcmgvm 22 xyxx vvvcvm 22

September 10, 2009

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 = D2.

= [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 kgg = 9.8; % Acceleration of gravity, in m/sdiam = 0.07; % Diameter of baseball, in mgamma = 0.25; % Coefficient of drag in air at STP, in Ns^2/m^2c = 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]);

September 10, 2009

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; endy = y(1:n-1,:);

pos = int_yp(T,V);

plot(19.3*T,23.0*T-4.9*T.^2); % Plot vacuum casehold onplot(pos(:,1),pos(:,2),'color','red'); % Overplot quadratic drag casehold off

September 10, 2009

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 theequivalent trajectory in a vacuum.

Note also that the baseballdoes not reach quite as highas in a vacuum, and reachesits peak earlier.

You will be given homeworkproblems in which I will askyou to try your hand at suchnumerical solutions and plotting.I will help you learn these very

useful skills, or you can make use of the Matlab helpers.

September 10, 2009

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:

BvF q

Bvv qm

),0,0( BB),,( zyx vvvv

)0,,( BvBv xy Bv

0

z

xy

yx

vm

Bqvvm

Bqvvm

September 10, 2009

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 = 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 as a vector in thecomplex plane whose components are vx and vy.

xy

yx

vv

vv

imaginarypart

real part

vx

vy = v x+ iv y

= vx + ivy

September 10, 2009

Motion of a Charge in a Uniform Magnetic Field-3

Next, we take the time derivative of :

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.

)( yxxyyx ivvivivviv

i

tiAe