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An Introduction to Rocket Science using STOMP, Scratch and GeoGebra Adrian Oldknow Some ideas for cross-curricular projects from KS2 to KS5 adrian@ccite.org July 2015

The classic STOMP air rocket is available in the UK from Amazon for c£8. You can also buy parts to make your own air rocket launchers from Mindsets Online, such as the Bottle Rocket and the Mindsets Rocket launcher kits. Or you could design and build your own from scratch. There are basically just two controls. You can adjust the launch angle using the butterfly nut, and the launch speed by the force you apply when you jump on the plastic box. For any launch you can measure the subsequent time of flight, using a stop watch. You can also measure the horizontal distance travelled, called the range, using a tape measure, trundle wheel or by pacing. So we have a simple experiment to be performed outdoors on a dry still day, or in the sports hall. Once the rocket leaves the launch tube the only forces acting on it are the gravitational pull of the Earth downwards, and, possibly, air resistance. The air rocket is a classic example of something called Projectile Motion, which was studied extensively by the Italian scientist and mathematician, Galileo Galilei, 1564 – 1642. Galileo had a long and varied career which would make a good topic for a project. He developed his own scientific instruments, especially the telescope, and made significant contributions to astronomy. He worked out the mathematical relationships between the launch speed V (metres per second), launch angle A (degrees), flight time T (seconds) and range R (metres) based on some simple assumptions. These are that:

there is no air resistance or wind

the ground is flat

the launch is from ground level

the acceleration of the rocket is constant (g metres per second per second) and downwards

the motion can be consider separately, as a horizontal component with no acceleration and a vertical component with constant acceleration g downwards.

In order to develop the basic ideas we can use a realistic example flight with very simple values for the variables. If we launch the rocket at A = 45° then the horizontal and vertical speeds are equal. If we assume they are each 10 mps (metres per second) then the actual launch speed V should be 10√2 ≈ 14.14 mps, which we can round to 14 mps. The usual value taken for the acceleration due to gravity downwards, g, is 9.82 mpsps, which we can round to 10 mpsps (metres per second per second). We can use the free GeoGebra software to illustrate the basic ideas. The software can be used on many different devices, at home or in school. You might find the Mathematics in Motion book on the National STEM Centre’s STEM Activities with TI-Nspire a useful starting point for ideas.

I am using the Graphics View to provide the `graph-paper’ on which we can create diagrams and enter text.

The aim is to try and understand how to interpret and use `constant acceleration’.

If the starting speed upwards is U = 10 mps, then after 1 second its upward speed will be 10 mps less, i.e. 0 mps. What does this mean in terms of the rocket’s flight?

After 2 seconds its upward speed will be another 10 mps less, i.e. -10 mps. So we are now into negative numbers! What does an upward speed of -10 mps mean? “What goes up must come down”!

It would be very useful to have a video clip of the STOMP rocket in flight which can be displayed e.g. using Windows Media Player. Using a right-click you can select `Enhancements’ and `Play speed settings’ to allow you to single-step the video clip.

The free Tracker software from the USA Open Source Physics programme also provides useful tools for video play-back, including single-stepping. The controls appear beneath the video display. You just need to use the `Video’ tab in the toolbar to tell Tracker where to find the video clip to be played. If you are going to take your own video clip it is advisable to mount the camera on a tripod. You need to be roughly in the centre of the scene of action and must not track or zoom the camera – keep it as still as possible. Preferably find a background against which the rocket will be easy to see, and include a feature whose dimensions you can measure, so you can extract accurate data from the video clip later. A frame round of 30 frames per second should be adequate.

So now back to the maths. After 1 second the STOMP rocket will be at the highest point of the flight. In that second its upward speed dropped steadily from 10 mps to 0 mps, so its average speed was 5 mps. So in 1 second it would have risen 5 metres vertically. In that time its sideways speed stayed constant at S = 10 mps, so it travelled 10 metres sideways.

Returning to GeoGebra we can bring in a Spreadsheet View, and introduce the idea of fractions of a second in decimals. I have also created a `slider’ for the value of the time T in seconds from 0 to 2 in steps of 0.1 seconds. The missing values for the Sideways speed will all be 10 mps, since S stays constant. The Upward speed U will decrease by 1 mps for every 1/10th of a second, so the missing values for U will be 9, 8, 7, 6, 5, 4, 3, 2, 1 mps. In each 1/10th of a second the rocket will travel 1m sideways so the missing values for X will be 1, 2, 3, 4, 5, 6, 7, 8, 9 metres. To work out the value of Y after 0.3 seconds, say we know it started with an upward speed of 10 mps and finished with a speed of 10 – 0.3 x 10 = 7 mps. So its average speed was (10 + 7)/2 = 8.5 mps, and so the upwards distance travelled was 8.5x0.3 = 2.55 m. The missing Y values are thus 0.95, 1.8, 2.55, 3.2, 3.75, 4.2, 4.55, 4.8, 4.95 metres.

From observation and the videos we know that if wind and air-resistance are negligible then the shape of the path of the rocket will be symmetrical. Galileo knew that this shape was a curve called a parabola, which ancient Greek mathematicians, such as Archimedes and Apollonius, had studied.

So we know that after 2 seconds the height of the rocket will be 0 metres and so it will have landed. In that time it will have travelled 10x2 = 20 metres sideways. So we know that the time of flight is 2 seconds and the range is 20 metres. In order to enter formulae into a spreadsheet, such as in GeoGebra we are already using a form of `algebra’ where we are using symbols like C4 to refer to cells which contain numbers.

The formula for cell C3 will look something like: C3 = C2 – 10*A3, where 10 is the value for the constant acceleration g mpsps. In order for this to copy down correctly to give the next formulae: C4 = C2 - 10*A4, C5 = C2 – 10*A5 etc. we need to distinguish between a changing (relative) and a non-changing (absolute) value of a cell. So the standard way of writing the formula to be copied is as C3 = $C$2 – 10*A3 . The corresponding formulae for X and Y could be: D3 = $B$2*A3 and E3 = ($C$2 + C3)/2*A3. These are used in the following Excel example.

The graph is a `scattergram’ using data from columns D and E to show the path of the rocket.

Personally I try to avoid the use of the dollar symbol “$” in defining absolute references to cells and to “Name” them instead. If you right click on a cell like F2 you can select “Define Name…”.

Cell A2 is named T Cell B2 is named S Cell C2 is named U Cell F2 is named gap Cell G2 is named g.

So now a formula in Excel can look very much like an expression in algebra. For example cell C3 now looks like: “C3 = U – g*A3”, and cell E3 looks like: “E3 = (U + C3)/2*A3”.

Mathematically the values in the columns of the spreadsheet are from three different types of function.

Column B, the sideways speed, is a constant – its graph against time is the blue horizontal line. Column D, the sideways distance travelled, goes up in equal steps – its graph against time is the green straight line. This is called a linear function. Column C, the upwards speed, goes down in equal steps – its graph is the red straight line. It is also a linear function. Column E, the height, goes up in decreasing steps, reaches a maximum and then comes down in increasing steps – its graph against time is the purple curve, called a parabola. It is called a quadratic function. If we insert a couple of extra columns between E and F we can study how the differences of column E behave. I have called the new column G the “1st diff” and the new column H the “2nd diff”.

The formula for cell F3 is: “F3 = E3 – E2” and that has been copied down as far as F22. The formula for G4 is: “G4 = F4 – F3” and that has been copied down as far as G9. So you can see that the 1st differences of a quadratic function are linear, and the 2nd differences are constant. This gives quite an efficient way to compute the next row of values.

G10 would be -0.10 F10 would be 0.35 + -0.10 = 0.25 E10 would be 4.55 + 0.25 = 4.80

and so on for each following row.

This technique was exploited by the British mathematician, Charles Babbage, 1791-1871 who designed the first, mechanical, computer in the 1820s – called the Difference Engine. Each turn of the crank computed the next row of values of the function and its differences. Each spindle of gears corresponds to a column in our modern spreadsheet.

Babbage went on to design more complicated computers which used programs on punched cards devised by Ada Lovelace 1815 – 1852. Fortunately computers and programming have come a long way since her day. We will first see how we can use variable in the Scratch programming language to compute the positions of our STOMP rocket, and then see how we can use these to control the graphics to simulate a STOMP flight.

The only changes between the Scratch program and the Spreadsheet version are the introduction of a variable called U0 to hold the starting value of the Upward speed U as 10 mps, and a `count’ variable. Use the Data commands to set up the variables and lists needed to produce the spreadsheet. The variable `count’ corresponds to each row of the sheet and the lists A to E correspond to the columns. To make the program more manageable I have defined a block of commands called “Set Up” to save all the starting values for the program. Once this been called just once by the program the main work is in the repeat loop which

computes each new value for each variable and stores them in the list. While it was not strictly necessary I also included a “wait” command in the loop so that it runs it so-called “real time”. Now we know the computations are working correctly we can use the X and Y values to place a picture (or “sprite”) of the rocket at some position on the screen.

The program is essentially the same, but I have edited the sprite to replace the `Scratch the dog’ image with a yellow rectangle and red circle to represent the rocket. The variables m, p and q have been introduced to move the origin to (-100,-50) for the start of the flight and to scale up distances from metres to pixels by a factor of 10. Now the flight is in real-time, but it is (a) rather lumpy (a steppy-graph) and (b) the rocket’s angle doesn’t change!

We can make the flight smoother by reducing the gap to a smaller time interval. We also need to change the direction angle A so that it starts at 45°, is 90° (sideways) at the half way point, and 135° at the end of the flight. For simplicity I have assumed for now that it changes in a linear fashion. You can now add any `bells and whistles’ that take your fancy. Maybe add some scenery, perhaps some sound? The projectile could be something else, like a golf-ball maybe. If you wanted to simulate a golf game you would need to be able to change the speed and angle of the shot. Fortunately that’s quite easy!

Now that we are using a computer to carry out the calculations we no longer need to restrict ourselves to the simple assumptions that the angle of launch was 45° and that the horizontal and vertical speeds S and U start by both being equal to 10 mps. If we launch with a speed of V mps at an angle A to the ground then the initial speeds will be S0 = V x cos A and U0 = V x sin A. As before, the rocket will stop climbing when its vertical velocity U becomes 0. If this is at time t then U = U0 – g x t = 0 and so t = U0/g. So the total time of flight T will be double this: T = 2U0/g. So the horizontal distance travelled will be R = S0 x T = 2 U0 x S0 / g. In our simple experiment we had U0 = S0 = 10 mps and g = 10 mpsps, giving R = 20m – which checks out. If we replace U0 with V x cos A and S0 with V x sin A the formula for the range becomes R = 2 V2/g sin A cos A. A bit of advanced trigonometry gives the formula: sin 2A = 2 sin A cos A – so R = V2/g sin 2A.

That’s the bit of maths which tells us that for any given initial speed V the launch angle A for which the rocket goes furthest will be 45°, since sin 2x45° = sin 90° = 1, which is the biggest value the sine of an angle can have. While not a proof, we can check the formula in GeoGebra’s spreadsheet, say. We can see that for each tabulated value of the angle A in column B, the formulae for columns E and G yield identical results. We can also see the result graphically by plotting the graphs of the two functions and seeing that they coincide. We can now go back to the Scratch simulation and make it work for any speed V and angle A.

In the new version the variable V holds the launch speed in mps, and A0 holds the launch angle. Scratch has built-in functions to compute sines and cosines, so we can use them to compute the initial sideways and upward speeds, S0 and U0. The variable T computes the total flight time, and A holds the direction angle for the rocket measured from the vertical. The variable a holds the amount this angle changes by in each small time interval, assuming that it changes at a constant rate. The variable t holds the current time in seconds.

We have pretty well developed all the theory we need to produce animated simulations of projectile motion. Returning to GeoGebra we can build an animated simulation which shows us how to model the angle of the rocket more accurately. The launch velocity V is entered as a constant in the Input bar as `V = 15’ and the launch angle A as `A = 60°’. Similarly `g = 9.8’. The initial horizontal speed as `S0 = V*cos(A)’ and vertical speed as `U0 = V*sin(A)’. The flight duration T is given by `T = 2*U0/g’, and a slider is set up for the time variable t from 0 to T in steps of 0.01. The speeds and coordinates are entered as functions of t: `S(t) = S0’, `U(t) = U0 – g*t’, `X(t) = S(t)*t’ and `Y(t) = (U0+U(t))/2*t’. The point R is defined as `R = (X(t),Y(t))’. As the slider for t is dragged right and left so the point R moves along a parabolic path. The point R can be selected to leave a `Trace’ which shows its successive positions. Using the 4th icon on the toolbar you can select to draw a `locus’; click on the point R and then on the slider point for t. The point R can also be selected to be `animated’. This produces a small Start/Pause button in the bottom left of the Graphics View. I have drawn horizontal and vertical lines through R to cut the axes. The variables `s = S(t)’ and `u = U(t)’ are used to define the red and blue vectors which show the current upward and sideways velocities of the rocket at any time, and tell us which way the rocket should be pointing! The angle α with the horizontal has been measured. We can compute the value for this as the angle whose tangent is u/s, using the function `β = arctan(u/s)’.

Again, now you have met some of the basic functions in GeoGebra you could build more exotic animations by creating shapes around the moveable point R. Technically we have developed what are called the parametric equations for the path of the rocket (X(t),Y(t)).

We can now return to the Scratch simulation and use the latest modification to create the correct heading for the rocket. To simulate motion like a tennis serve you will need to find out what adjustments are needed for a launch from above ground level. For a more realistic simulation you might see what you can find out about air-resistance.