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 January 17th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

Numerical and Analytical Solutions 2: Constant Acceleration

November 3, 2016

Previously, I showed how to solve a simple problem of motion at a constant velocity analytically and numerically. Because of the nature of the problem both solutions gave the same result. Now we’ll try a constant acceleration problem which should highlight some of the key differences between the two approaches, particularly the tradeoffs you must make when using numerical approaches.

The Problem

  • A ball starts at the origin and moves horizontally with an acceleration of 0.2 m/s2. Print out a table of the ball’s position (in x) with time (every second) for the first 20 seconds.

Analytical Solution
We know that acceleration (a) is the change in velocity with time (t):

a = \frac{dv}{dt}

so if we integrate acceleration we can find the velocity. Then, as we saw before, velocity (v) is the change in position with time:

v = \frac{dx}{dt}

which can be integrated to find the position (x) as a function of time.

So, to summarize, to find position as a function of time given only an acceleration, we need to integrate twice: first to get velocity then to get x.

For this problem where the acceleration is a constant 0.2 m/s2 we start with acceleration:

\frac{dv}{dt} = 0.2

which integrates to give the general solution,

v = 0.2 t + c

To find the constant of integration we refer to the original question which does not say anything about velocity, so we assume that the initial velocity was 0: i.e.:

at t = 0 we have v = 0;

which we can substitute into the velocity equation to find that, for this problem, c is zero:

v = 0.2 t + c
0 = 0.2 (0) + c
0 = c

making the specific velocity equation:
v = 0.2 t

we replace v with dx/dt and integrate:

\frac{dx}{dt} = 0.2 t
x = \frac{0.2 t^2}{2} + c
x = 0.1 t^2 + c

This constant of integration can be found since we know that the ball starts at the origin so

at t = 0 we have x = 0, so;

x = 0.1 t^2 + c
0 = 0.1 (0)^2 + c
0 = c

Therefore our final equation for x is:

x = 0.1 t^2

Summarizing the Analytical

To summarize the analytical solution:

a = 0.2
v = 0.2 t
x = 0.1 t^2

These are all a function of time so it might be more proper to write them as:

a(t) = 0.2
v(t) = 0.2 t
x(t) = 0.1 t^2

Velocity and acceleration represent rates of change which so we could also write these equations as:

a(t) = \frac{dv}{dt} = 0.2
v(t) = \frac{dx}{dt} = 0.2 t
x(t) = x = 0.1 t^2

or we could even write acceleration as the second differential of the position:

a(t) = \frac{d^2x}{dt^2} = 0.2
v(t) = \frac{dx}{dt} = 0.2 t
x(t) = x = 0.1 t^2

or, if we preferred, we could even write it in prime notation for the differentials:

a(t) = x
v(t) = x
x(t) = x(t) =0.1 t^2

The Numerical Solution

As we saw before we can determine the position of a moving object if we know its old position (xold) and how much that position has changed (dx).

x_{new} = x_{old} + dx

where the change in position is determined from the fact that velocity (v) is the change in position with time (dx/dt):

v = \frac{dx}{dt}

which rearranges to:

dx = v dt

So to find the new position of an object across a timestep we need two equations:

dx = v dt
x_{new} = x_{old} + dx

In this problem we don’t yet have the velocity because it changes with time, but we could use the exact same logic to find velocity since acceleration (a) is the change in velocity with time (dv/dt):

a = \frac{dv}{dt}

which rearranges to:

dv = a dt

and knowing the change in velocity (dv) we can find the velocity using:

v_{new} = v_{old} + dv

Therefore, we have four equations to find the position of an accelerating object (note that in the third equation I’ve replaced v with vnew which is calculated in the second equation):

dv = a dt
v_{new} = v_{old} + dv
dx = v_{new} dt
x_{new} = x_{old} + dx

These we can plug into a python program just so:

motion-01-both.py

from visual import *

# Initialize
x = 0.0
v = 0.0
a = 0.2
dt = 1.0

# Time loop
for t in arange(dt, 20+dt, dt):

     # Analytical solution
     x_a = 0.1 * t**2

     # Numerical solution
     dv = a * dt
     v = v + dv
     dx = v * dt
     x = x + dx

     # Output
     print t, x_a, x

which give output of:

>>> 
1.0 0.1 0.2
2.0 0.4 0.6
3.0 0.9 1.2
4.0 1.6 2.0
5.0 2.5 3.0
6.0 3.6 4.2
7.0 4.9 5.6
8.0 6.4 7.2
9.0 8.1 9.0
10.0 10.0 11.0
11.0 12.1 13.2
12.0 14.4 15.6
13.0 16.9 18.2
14.0 19.6 21.0
15.0 22.5 24.0
16.0 25.6 27.2
17.0 28.9 30.6
18.0 32.4 34.2
19.0 36.1 38.0
20.0 40.0 42.0

Here, unlike the case with constant velocity, the two methods give slightly different results. The analytical solution is the correct one, so we’ll use it for reference. The numerical solution is off because it does not fully account for the continuous nature of the acceleration: we update the velocity ever timestep (every 1 second), so the velocity changes in chunks.

To get a better result we can reduce the timestep. Using dt = 0.1 gives final results of:

18.8 35.344 35.532
18.9 35.721 35.91
19.0 36.1 36.29
19.1 36.481 36.672
19.2 36.864 37.056
19.3 37.249 37.442
19.4 37.636 37.83
19.5 38.025 38.22
19.6 38.416 38.612
19.7 38.809 39.006
19.8 39.204 39.402
19.9 39.601 39.8
20.0 40.0 40.2

which is much closer, but requires a bit more runtime on the computer. And this is the key tradeoff with numerical solutions: greater accuracy requires smaller timesteps which results in longer runtimes on the computer.

Post Script

To generate a graph of the data use the code:

from visual import *
from visual.graph import *

# Initialize
x = 0.0
v = 0.0
a = 0.2
dt = 1.0

analyticCurve = gcurve(color=color.red)
numericCurve = gcurve(color=color.yellow)
# Time loop
for t in arange(dt, 20+dt, dt):

     # Analytical solution
     x_a = 0.1 * t**2

     # Numerical solution
     dv = a * dt
     v = v + dv
     dx = v * dt
     x = x + dx

     # Output
     print t, x_a, x
     analyticCurve.plot(pos=(t, x_a))
     numericCurve.plot(pos=(t,x))

which gives:

Comparison of numerical and analytical solutions using a timestep (dt) of 1.0 seconds.

Comparison of numerical and analytical solutions using a timestep (dt) of 1.0 seconds.

Citing this post: Urbano, L., 2016. Numerical and Analytical Solutions 2: Constant Acceleration, Retrieved January 17th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

Nukemap

March 8, 2016

“If I were to convert all of my body, my mass to energy, how much could I blow up?”

The Nukemap website tries to help answer that question.

From the Nukemap website.

From the Nukemap website.

However, if we use the equation E = mc2, we can convert a 50 kg student into the explosive energy of the equivalent of 1,000,000 kilotons of TNT, which the newer website can’t quite handle.

Citing this post: Urbano, L., 2016. Nukemap, Retrieved January 17th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

Physics: Theories of Everything (Mapped)

March 7, 2016

An excellent overview of the multitude of active theories and hypotheses–like quantum gravity, string theory–physicists are investigating to try to explain the universe.

From Theories on Everything Mapped

From Theories on Everything Mapped

In the quest for a unified, coherent description of all of nature — a “theory of everything” — physicists have unearthed the taproots linking ever more disparate phenomena. With the law of universal gravitation, Isaac Newton wedded the fall of an apple to the orbits of the planets. Albert Einstein, in his theory of relativity, wove space and time into a single fabric, and showed how apples and planets fall along the fabric’s curves. And today, all known elementary particles plug neatly into a mathematical structure called the Standard Model. But our physical theories remain riddled with disunions, holes and inconsistencies. These are the deep questions that must be answered in pursuit of the theory of everything.

–Natalie Wolchover in Theories of Everything Mapped on Quanta Magazine.

Citing this post: Urbano, L., 2016. Physics: Theories of Everything (Mapped), Retrieved January 17th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

Interactive Electric Fields (with Paper.js)

February 26, 2016

Drag the charges around.

The force field created by the interaction of two electric charges (one positive and one negative). The source is at http://soriki.com/fields/electric/.

Citing this post: Urbano, L., 2016. Interactive Electric Fields (with Paper.js), Retrieved January 17th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

How to Cool Something to a Billionth of a Kelvin

January 10, 2016

How to create ultra-cold temperatures (and what it tells us about the universe).

And this article (Below Absolute Zero: Negative Temperatures Explained) tells how to get below absolute zero.

Citing this post: Urbano, L., 2016. How to Cool Something to a Billionth of a Kelvin, Retrieved January 17th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

Circuit Basics

December 30, 2015

Studying voltage and current in circuits can start with two laws of conservation.

  • KCL: Current flow into a node must equal the flow out of the node. (A node is a point on the wire connecting components in a circuit–usually a junction).
(KCL: Kirchoff's Current Law) Current flowing into any point on a circuit is equal to the current flowing out of it, A simple circuit with a voltage source (like a battery) and a resistor.

(KCL: Kirchoff’s Current Law) Current flowing into any point on a circuit is equal to the current flowing out of it, A simple circuit with a voltage source (like a battery) and a resistor.

  • KVC: The sum of all the voltage differences in a closed loop is zero.

KVL: The voltage difference across the battery (9 Volts) plus the voltage difference across the resistor (-9 Volts) is equal to zero.

KVL: The voltage difference across the battery (9 Volts) plus the voltage difference across the resistor (-9 Volts) is equal to zero.

Things get more interesting when we get away from simple circuits.

Current flow into a node (10 A) equals the flow out of the node (7 A + 3 A).

Current flow into a node (10 A) equals the flow out of the node (7 A + 3 A).

Note that the convention for drawing diagrams is that the current move from positive (+) to negative (-) terminals in a battery. This is opposite the actual flow of electrons in a typical wired circuit because the current is a measure of the movement of negatively charged electrons, but is used for historical reasons.

Based on the MIT OpenCourseWare Introduction to Electrical Engineering and Computer Science I Circuits 6.01SC Introduction to Electrical Engineering and Computer Science Spring 2011.

Citing this post: Urbano, L., 2015. Circuit Basics, Retrieved January 17th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

New Particles

December 16, 2015

Physicists in Europe Find Tantalizing Hints of a Mysterious New Particle: This new particle, if confirmed to exist (the data is not conclusive) seems to go beyond the Standard Model of physics that we know and love.

The last sub-atomic particle discovered was the Higgs boson, which is shown in the graph below.

Finding the Higgs Boson "The strongest evidence for this new particle comes from analysis of events containing two photons. The smooth dotted line traces the measured background from known processes. The solid line traces a statistical fit to the signal plus background. The new particle appears as the excess around 126.5 GeV. The full analysis concludes that the probability of such a peak is three chances in a million. " from ATLAS.

Finding the Higgs Boson “The strongest evidence for this new particle comes from analysis of events containing two photons. The smooth dotted line traces the measured background from known processes. The solid line traces a statistical fit to the signal plus background. The new particle appears as the excess around 126.5 GeV. The full analysis concludes that the probability of such a peak is three chances in a million. ” from ATLAS.

Citing this post: Urbano, L., 2015. New Particles, Retrieved January 17th, 2017, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: Montessori Muddle; Hat tip: Montessori Muddle.

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