Projectile Paths

January 10, 2017

Paths of a projectile. The half oval represents a line connecting the maximum heights of each projectile.

Paths of a projectile.

I had my Numerical Methods student calculate the angle that would give a ballistic projectile its maximum range, then I had them write a program that did the the same by just trying a bunch of different angles. The diagram above is what they came up with.

It made an interesting pattern that I converted into a face-plate cover for a light switch that I made using the laser at the TechShop.

Face plate cover.

Face plate cover.

Citing this post: Urbano, L., 2017. Projectile Paths, Retrieved June 27th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

Maximum Range of a Potato Gun

December 18, 2016

One of the middle schoolers built a potato gun for his math class. He was looking a the mathematical relationship between the amount of fuel (hair spray) and the hang-time of the potato. To augment this work, I had my Numerical Methods class do the math and create analytical and numerical models of the projectile motion.

One of the things my students had to figure out was what angle would give the maximum range of the projectile? You can figure this out analytically by finding the function for how the horizontal distance (x) changes as the angle (theta) changes (i.e. x(theta)) and then finding the maximum of the function.

Initial velocity vector (v) and its component vectors in the x and y directions.

Initial velocity vector (v) and its component vectors in the x and y directions for a given angle.

Distance as a function of the angle

In a nutshell, to find the distance traveled by the potato we break its initial velocity into its x and y components (vx and vy), use the y component to find the flight time of the projectile (tf), and then use the vx component to find the distance traveled over the flight time.

Starting with the diagram above we can separate the initial velocity of the potato into its two components using basic trigonometry:

\cos{\theta} = \frac{v_x}{v}
\sin{\theta} = \frac{v_y}{v} ,

so,

v_x = v \cos{\theta} ,
v_y = v \sin{\theta}

Now we know that the height of a projectile (y) is given by the function:

 y(t) = \frac{a t^2}{2} + v_0 t + y_0

(you can figure this out by assuming that the acceleration due to gravity (a) is constant and acceleration is the second differential of position with respect to time.)

To find the flight time we assume we’re starting with an initial height of zero (y0 = 0), and that the flight ends when the potato hits the ground which is also at zero ((yt = 0), so:

 0 = \frac{a t^2}{2} + v_0 t + 0

 0 = \frac{a t^2}{2} + v_0 t

Factoring out t gives:

 0 = t ( \frac{a t}{2} + v_0)

Looking at the two factors, we can now see that there are two solutions to this problem, which should not be too much of a surprise since the height equation is parabolic (a second order polynomial). The solutions are when:

 t = 0

  \frac{a t}{2} + v_0 = 0

The first solution is obviously the initial launch time, while the second is going to be the flight time (tf).

  \frac{a t_f}{2} + v_0 = 0

  t_f = - \frac{2 v_0}{a}

You might think it’s odd to have a negative in the equation, but remember, the acceleration is negative so it’ll cancel out.

Now since we’re working with the y component of the velocity vector, the initial velocity in this equation (v0) is really just vy:

   v_0 = v_y

so we can substitute in the trig function for vy to get:

  t_f = - \frac{2 v  \sin{\theta}}{a}

Our horizontal distance is simply given by the velocity in the x direction (vx) times the flight time:

  x = v_x t_f

which becomes:

  x = v_x  \left(- \frac{2 v  \sin{\theta}}{a}\right)

and substituting in the trig function for vx (just to make things look more complicated):

  x = \left(  v \cos{\theta} \right)  \left(- \frac{2 v  \sin{\theta}}{a}\right)

and factoring out some of the constants gives:

  x = -\frac{v^2}{a} 2 \sin{\theta}\cos{theta}

Now we have distance as a function of the launch angle.

We can simplify this a little by using the double-angle formula:

  \sin{2\theta} = 2 \sin{\theta}\cos{theta}

to get:

  x = -\frac{v^2}{a} \sin{2\theta}

Finding the maximum distance

How do we find the maxima for this function. Sketching the curve should be easy enough, but because we know a little calculus we know that the maximum will occur when the first differential is equal to zero. So we differentiate with respect to the angle to get:

  \frac{dx}{d\theta} = -\frac{v^2}{a} 2 \cos{2\theta}

and set the differential equal to zero:

  0 = -\frac{v^2}{a} 2 \cos{2\theta}

and solve to get:

  \cos{2\theta}  = 0

  2\theta  = \cos^{-1}{(0)}

Since we remember that the arccosine of 0 is 90 degrees:

  2\theta  = 90^{\circ}

  \theta  = 45^{\circ}

And thus we’ve found the angle that gives the maximum launch distance for a potato gun.

Citing this post: Urbano, L., 2016. Maximum Range of a Potato Gun, Retrieved June 27th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

The Joy of Math

February 2, 2016

Regardless, if learning is to be as efficient and deep as possible, it’s essential that it be done freely. That means giving children a voice in which activities to participate, for how long, and also the level of mastery they want to achieve. (“This is the biggest clash with traditional curriculum development,” Droujkova notes.)
— Vangelova (2014): 5-Year-Olds Can Learn Calculus

This article provides a lot of evidence to support the notion that the conceptual aspects of calculus and other “higher level” forms of math should be taught at all age levels, not just at the end of high school or in college.

The presentation below elaborates:

Citing this post: Urbano, L., 2016. The Joy of Math, Retrieved June 27th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

Dilation

December 14, 2015

Dilation (scaling) of a quadrilateral by 2x.

Dilation (scaling) of a quadrilateral by 2x.

A quick program that animates scaling (dilation) of shapes by scaling the coordinates. You type in the dilation factor.

dilation.py

from visual import *

#axes
xmin = -10.
xmax = 10.
ymin = -10.
ymax = 10.
xaxis = curve(pos=[(xmin,0),(xmax,0)])
yaxis = curve(pos=[(0,ymin),(0,ymax)])

#tick marks
tic_dx = 1.0
tic_h = .5
for i in arange(xmin,xmax+tic_dx,tic_dx):
    tic = curve(pos=[(i,-0.5*tic_h),(i,0.5*tic_h)])
for i in arange(ymin,ymax+tic_dx,tic_dx):
    tic = curve(pos=[(-0.5*tic_h,i),(0.5*tic_h,i)])

#stop scene from zooming out too far when the curve is drawn
scene.autoscale = False

# define curve here
shape = curve(pos=[(-1,2), (5,3), (4,-1), (-1,-1)])

shape.append(pos=shape.pos[0])
shape.color = color.yellow
shape.radius = 0.1
shape.visible = True

#dilated shape
dshape = curve(color=color.green, radius=shape.radius*0.9)
for i in shape.pos:
    dshape.append(pos=i)

#label
note = label(pos=(5,-8),text="Dilation: 1.0", box=False)
intext = label(pos=(5,-9),text="> x", box=False)

#scaling lines
l_scaling = False
slines = []
for i in range(len(shape.pos)):
    slines.append(curve(radius=shape.radius*.5,color=color.red, pos=[shape.pos[i],shape.pos[i],shape.pos[i]]))


#animation parameters
animation_time = 1. #seconds
animation_smootheness = 30
animation_rate = animation_smootheness / animation_time

x = ""
while 1:
    #x = raw_input("Enter Dilation: ")
    if scene.kb.keys: # event waiting to be processed?
        s = scene.kb.getkey() # get keyboard info

        #print s
        if s <> '\n':
            x += s
            intext.text = "> x "+x
        else:
            try:
                xfloat = float(x)
                note.text = "Dilation: " + x

                endpoints = []
                dp = []
                for i in shape.pos:
                    endpoints.append(float(x) * i)
                    dp.append((endpoints[-1]-i)/animation_smootheness)
                #print "endpoints: ", endpoints
                #print "dp:        ", dp
                for i in range(animation_smootheness):
                    for j in range(len(dshape.pos)):
                        dshape.pos[j] = i*dp[j]+shape.pos[j]
                    rate(animation_smootheness)
                    if slines:
                        for i in range(len(shape.pos)):
                            slines[i].pos[1] = vector(0,0)
                            slines[i].pos[-1] = dshape.pos[i]

                for i in range(len(shape.pos)):
                    dshape.pos[i] = endpoints[i]
                    slines[i].pos[-1] = dshape.pos[i]

                for i in range(len(shape.pos)-1):
                    print shape.pos[i], "--->", dshape.pos[i]
                    
            except:
                #print "FAIL"
                failed = True
                intext.text = "> x "
            x = ""
            
                   

Citing this post: Urbano, L., 2015. Dilation, Retrieved June 27th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

Toothpick Shapes’ Sequences

August 27, 2015

Toothpick shape sequences.

Toothpick shape sequences.

Using a sequence of connected shapes to introduce algebra and graphing to pre-Algebra students.

Make a geometric shape–a square perhaps–out of toothpicks. Count the sides–4 for a square. Now add another square, attached to the first. You should now have 7 toothpicks. Keep adding shapes in a line and counting toothpicks. Now you can:

  • make a table of shapes versus toothpicks,
  • write the sequence as an algebraic expression
  • graph the number of shapes versus the number of toothpicks (it should be a straight line),
  • figure out that the increment of the sequence–3 for a square–is the slope of the line.
  • show that the intercept of the line is when there are zero shapes.

Then I had my students set up a spreadsheet where they could enter the number of shapes and it would give the number of toothpicks needed. Writing a small program to do the same works is the next step.

Citing this post: Urbano, L., 2015. Toothpick Shapes' Sequences, Retrieved June 27th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

Math Gifs

September 2, 2014

Mike Schmidt passed along this link to Lisa Winter’s post collecting 21 GIFs That Explain Mathematical Concepts.

For example:

Converting a trigonometric function (sin curve) from Cartesian to polar coordinates.

Converting a trigonometric function (sin curve) from Cartesian to polar coordinates. Source: “Cartesian to polar” by KieffOwn work. Licensed under Public domain via Wikimedia Commons.

Citing this post: Urbano, L., 2014. Math Gifs, Retrieved June 27th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

Project Euler: Math/Programming Challenge

February 26, 2014

On the recommendation of Mr. Schmidt, two of my students have been quite fascinated over the last few days trying to solve problems on Project Euler. They’ve been working on them together to, I suspect, the detriment of some of their other classes, but as their math teacher I find it hard to object.

An example problem is something like this:

If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is 23.

Find the sum of all the multiples of 3 or 5 below 1000.

They’ve been solving them numerically using Python. It’s been quite fascinating to see.

Citing this post: Urbano, L., 2014. Project Euler: Math/Programming Challenge, Retrieved June 27th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

Sine Curves

February 18, 2014

Frequency.

Frequency.


I’ve slapped together this simple VPython program to introduce sinusoidal functions to my pre-Calculus students.

  • Left and right arrow keys increase and decrease the frequency;
  • Up and down arrow keys increase and decrease the amplitude;
  • “a” and “s” keys increase and decrease the phase.
Amplitude.

Amplitude.

The specific functions shown on the graph are based on the general function:

y =  A \sin{F x + P}

where:

  • A — amplitude
  • F — frequency
  • P — phase
Phase. Note how the curve seems to move backward when the phase increases.

Phase. Note how the curve seems to move backward when the phase increases.

When I first introduce sinusoidal functions to my pre-Calculus students I have them make tables of the functions (from -2π to 2π with an interval of π/8) and then plot the functions. Then I’ll have them draw sets of sine functions so they can observe different frequencies, amplitudes, and phases.

from visual import *

class sin_func:
    def __init__(self, x, amp=1., freq=1., phase=0.0):
        self.x = x
        self.amp = amp
        self.freq = freq
        self.phase = phase

        self.curve = curve(color=color.red, x=self.x, y=self.f(x), radius=0.05)
        self.label = label(pos=(xmin/2.0,ymin), text="Hi",box=False, height=30)
        
    def f(self, x):
        y = self.amp * sin(self.freq*x+self.phase)
        return y

    def update(self, amp, freq, phase):
        self.amp = amp
        self.freq = freq
        self.phase = phase
        self.curve.y = self.f(x)
        self.label.text = self.get_eqn()

    def get_eqn(self):
        if self.phase == 0.0:
            tphase = ""
        elif (self.phase > 0):
            tphase = u" + %i\u03C0/8" % int(self.phase*8.0/pi)
        else:
            tphase = u" - %i\u03C0/8" % int(abs(self.phase*8.0/pi))
        print self.phase*8.0/pi

        txt = "y = %ssin(%sx %s)" % (simplify_num(self.amp), simplify_num(self.freq), tphase)
        return txt
    
def simplify_num(num):
    if (num == 1):
        snum = ""
    elif (num == -1):
        snum = "-"
    else:
        snum = str(num).split(".")[0]+" "
    return snum
        
amp = 1.0
freq = 1.0

damp = 1.0
dfreq = 1.0

phase = 0.0
dphase = pi/8.0

xmin = -2*pi
xmax = 2*pi
dx = 0.1

ymin = -3
ymax = 3

scene.width=640
scene.height=480

xaxis = curve(pos=[(xmin,0),(xmax,0)])
yaxis = curve(pos=[(0,ymin),(0,ymax)])

x = arange(xmin, xmax, dx)
#y = f(x)

func = sin_func(x=x)
func.update(amp, freq, phase)

while 1: #theta <= 2*pi:
    rate(60)

    if scene.kb.keys: # is there an event waiting to be processed?
        s = scene.kb.getkey() # obtain keyboard information
        #print s
        if s == "up":
            amp += damp
        if s == "down":
            amp -= damp
        if s == "right":
            freq += dfreq
        if s == "left":
            freq -= dfreq

        if s == "s":
            phase += dphase
        if s == "a":
            phase -= dphase

        func.update(amp, freq, phase)
        #update_curve(func, y)
        

Citing this post: Urbano, L., 2014. Sine Curves, Retrieved June 27th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

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