THE MATH BEHIND THE MACHINE The catapult is our culminating first-semester physics project. With it, we are able to apply many of the skills and concepts learned throughout the semester. EX: our catapult uses a spring to provide it with energy to fire a projectile. This spring contains elastic potential energy (EPE) which is converted into the translational kinetic energy (TKE) and gravitational potential energy (GPE) of a marble projectile when the catapult fires, both forms of energy we studied in Chapter 8. The marble the catapult fires behaves according to the projectile motion principles that we studied in Chapter 3. However, we also learned that projectile motion is a combination of two simpler motions which are vertical motion and horizontal motion that we studied in Chapter. Finally, to design, build, and fire the catapult effectively we will use the skills of science which we learned in Chapter 1. There are some quantities that we will measure in our catapult and there are others that we will calculate. Each group of students will measure the following using rulers, protractors, scales: l, the length of their catapult throwing arm (in m) k, the spring constant of their spring (in N/m), the launch angle of the catapult (in degrees), h, the launch height of your marble (in m) m, the mass of the marble projectile (in kg) These measurements, when used in conjunction with some formulas we have studied this semester will allow us to calculate the initial velocity, v o (in m/s), at which the catapult fires, the distance, x (in m), that the marble will travel before hitting the ground the height (in m)of the marble projectile y(x) at any moment in the marble s flight, the total elastic potential energy (EPE = ½kx ) the catapult contains when it is ready to fire, the kinetic energy (TKE = ½mv o ) of the marble when fired, its gravitational potential energy (GPE = mgh) of the marble when fired, the remaining elastic potential energy (EPE = ½kx ) of the spring after the catapult is fired. MEASURING. To measure the length of the catapult throwing arm, l, and the launch height, h, of the marble, you will use a simple meter stick and convert your units to meters. The launch angle,, is found with the protractor built into the catapult. The mass of the marble projectile, m, is 0.005 kg. The spring constant, k, of your braided rope catapult is 000 N/m. All these values are quantities that are measured. length, l, in meters mass of marble, m, in kgs spring constant, k, in Newtons/meter angle,, in degrees launch height, h, in meters 1
CALCULATING. ENERGY. In Chapter 8 we looked at energy considerations. Consider that there are three important energy states that the catapult is in: DRAW: DRAW: DRAW: 1 st STATE. Energy = 0. Catapult is in a relaxed position. nd STATE. E in =. Catapult is pulled back by the Work done by the hand which contained CPE (Chemical Potential Energy). The work done has given the catapult (form of energy). 3 rd STATE. E out =. Catapult fires. The chain stops it before the spring completely releases so the catapult still has some (form of energy). The marble has gained height so it has (form of energy). The marble is moving so the catapult also has (form of energy). The energy formula for the catapult is summarized as follows: E in = E out = With the formulas plugged in: = + Solve the formula above for v o, your hediondo (without plugging in numbers): v o = Now, given the initial conditions from before that you measured, find the initial velocity of the marble that is projected from your catapult. (Remember: the mass of the marble projectile, m, is 0.005 kg. The spring constant, k, of your braided rope catapult is 000 N/m). Assume the braided rope is stretched x = 0.005 m. v o = m/s
--------------------------------------------------------------------------- SIMPLE MACHINES. In Chapter 8 we looked at simple machines. A catapult is a not-so-simple machine that involves several simple machines. MULTIPLE CHOICE QUESTIONS. Choose the letter of the best answer. 1. The catapult is mainly what kind of simple machine? a. wheel & axle d. lever b. screw e. pulley c. wedge f. inclined plane. When we built the base of the catapult, to ensure that it was strong we used instead of. a. screws, nails d. wedges, screws b. nails, screws e. both a. and c. are correct c. screws, wedges f. both b. and d. are correct 3. Screws are simple machines themselves but at the top end of every screw there is almost always a and at the tip of every screw there is almost always a. a. wheel & axle, wedge d. lever, inclined plane b. wheel & axle, inclined plane e. wheel & axle, inclined plane c. inclined plane, wedge f. inclined plane, lever 4. To twist and tighten the rope that would become our spring we used a. a. wheel & axle d. lever b. screw e. pulley c. wedge f. inclined plane 5. The fulcrum of the lever that is in the catapult is where. a. the protractor is attached to the base of the catapult b. the rope braid wraps around the throwing arm c. the bottle cap is attached to the throwing arm d. the rope braid is attached to the base of the catapult 3
------------------------------------------------------------------------------ MOTION FORMULAS. In Chapter & 3 we looked at motion formulas. The energy formulas from above permit you to get the initial velocity, v o (in m/s), at which the marble projectile will be fired. Once the marble fires, a different formula is used to find how far (x) in meters, the marble will go. The formula you will use to find this is related to the formulas we used in Quarter 1 for vertical and horizontal motion. The catapult formula is as follows: y y o (tan v 4.9 (cos ) ) x x. MATCHING. Although the catapult formula looks complicated initially, we will find that it is an appearance only. Let s take apart this formula and conquer it. Match each variable from the formula on the left with its definition on the right: o ) 1. y. y o 3. x 4. 5. v o a. initial height of the marble (projectile), in m b. initial velocity of the marble (projectile), in m/s c. launch angle, in degrees d. horizontal distance the projectile moves, in m. e. height of the projectile at any moment, in m MULTIPLE CHOICE QUESTIONS. Choose the letter of the best answer. 1. The catapult formula looks most like which of the formulas we worked with in Quarter 1? a. x = v xo t d. y = y o + v yo t ½gt b. x = v xo t + ½at e. y = y o ½gt c. x = v xo t ½at. One BIG difference between the catapult formula and the one we worked with in Quarter 1 is that one particular quantity is missing from the CATAPULT formula which is: a. initial velocity d. gravity b. initial height e. time elapsed c. height thrown 3. Another BIG difference between the catapult formula and the one we worked with in Quarter 1 is that the quantity from Quest. Nr. has been replaced by what quantity? a. initial height d. initial velocity b. horizontal distance the projectile moves e. gravity c. time elapsed 4. The number 4.9 that appears in the catapult formula comes from: a. the spring constant d. ½ gravity (½g) b. the initial height e. it is not clear just by looking c. the initial velocity 4
4.9 Look closely at this catapult formula: y yo (tan ) x x v (cos ) ) o Simplified, the catapult formula actually looks like this: y = y o + ax bx where a = tan and b = 5 v o 4.9 (cos ) 5. At first the catapult formula looks like a lot of mumbo-jumbo but if we look at it carefully, we can see that it is simply the formula of a. a. line b. parabola c. circle d. ellipse e. hyperbola -------------------------------------------------------------------------------------------------------------- Realize that you will measure your launch angle with the protractor and you will have found the initial velocity v o from the energy formula as shown above so that a and b are just numbers that you will put in front of x and x in the formula. EXAMPLE: Suppose that the launch angle of your catapult = 40 and the initial height, y o, of the launch arm is 0.5 m, and your initial velocity,v o, that you got from your energy formulas was v o =.6 m/s. Plug in your values into your catapult formula and you get: 4.9 y 0.5 (tan 40) x x.6 (cos 40) ) 4.9 y 0.5 (0.839099631) x x 5.1 (0.7660444431) ) 4.9 y 0.5 (0.839099631) x x 5.1(0.586840888)) 4.9 y 0.5 (0.839099631) x x.9980853 y 0.5 (0.839099631) x 1.6376103x y = 0.5 + 0.839099631x 1.6376103x Graph the formula on your graphing calculator. Draw what it looks like here at right. a. Looking at the formula, how do you know that the parabola opens downward? b. Label the y-intercept on your graph (x, y). The y-intercept represents the the marble is thrown at. c. What does the vertex of the parabola represent in real life in terms of our catapult? d. What does the positive x-intercept of the parabola represent in real life in terms of our catapult?. ).
FINDING THE MAXIMUM HEIGHT OF THE MARBLE. To find the maximum height, graph the catapult formula in your graphing calculator, select CALC, and find the maximum. EX: Using the same values as before, we would graph the following. and we would get the following graph: Plot 1 Plot Plot 3 \Y 1 = 0.5 + xtan (40) 4.9x /((.6 *(cos(40)) ) \Y = \Y 3 = \Y = nd TRACE To find the maximum, hit CALC which is and choose 3: maximum, which, in this case is y =. Draw the coordinates of the maximum height (x, y) as they appear on your calculator screen on the graph at left FINDING HOW FAR THE MARBLE WILL GO. To find the distance the marble should go, graph the catapult formula in your graphing calculator, select CALC, and find the zero. EX: Using the same values as before, we would graph the following. Plot 1 Plot Plot 3 \Y 1 = 0.5 + xtan (40) 4.9x /((.6 *(cos(40)) ) \Y = \Y 3 = \Y = To find the maximum, hit CALC which is nd and choose : zero. Find the zero of your parabola which, in this case is x =. TRACE Draw the coordinates of the horizontal distance travelled (x, y) as they appear on your calculator screen on the graph at left This is theoretically how far (in meters) your catapult should throw when you shoot it the first time. THE END 6