Lab #6 – Energy of a Bouncing Ball GOALS In this lab you will learn: • How to use the momentum principle to solve for the initial velocity of a bouncing golf ball. • How to use the initial velocity of a bouncing golf ball and the energy principle to predict the maximum height of the ball, which you will compare with the height you measure. PART I You need a golf ball, a stopwatch, and a meter stick. The experiment requires at least 2 people: • One person holds the ball as high as possible and drops it. Record the initial height, yinitial, from which the ball is dropped. Doing so will maximize the time of flight of the ball and make it easier to measure the time accurately. This person will try to mark how high above the floor, ymax, the ball bounces on the first bounce. Use the bottom of the ball as your measurement point! • Another person starts the stopwatch when the ball first strikes the floor and stops it when the ball strikes the floor again. This is twice the time required to go from the floor to the maximum height after the bounce (in the approximation that we neglect air resistance). Trial 1 Δtfloortofloor = Δtfloortomax = yinitial = ymax = Trial 2 Δtfloortofloor = Δtfloortomax = yinitial = ymax = Using the momentum principle, use your experimental observations to determine the initial speed of the ball just after it bounces off the floor in the box below, by answering the following questions: Use m for the mass of the ball, v for the magnitude of the vertical component of the initial velocity, and ∆t for the time to reach the maximum height after the bounce off the floor. Be careful of signs. Write down in your notebook that you are using the ball as the system. SHOW ALL STEPS 1.) Draw a picture of the ball’s motion, clearly identifying the “initial” and “final” times. Also, draw a set of coordinate axes. 2.) Determine the following values symbolically. Do not plug in any numbers until later, though 0’s may be plugged in now. Initial momentum (just after bounce): �! = _______________, _______________, _____________ Final momentum (at top): �! = _______________, _______________, _____________ Force exerted on the ball by the Earth: �!" = _______________, _______________, _____________ Net force exerted on the ball: �!"# = _______________, _______________, _____________ 3.) Write the momentum principle, DO NOT PLUG IN ANYTHING YET: 4.) Using your expressions for �! , �!, and �!"# from above, solve for the initial speed in the vertical direction. DO NOT PLUG IN NUMBERS YET. Solve starting from Momentum prinCiple. 5.) Now plug in your experimental numbers to predict the initial speeds for your two trials. Show your calculations. (trial 1) initial v = _______________ m/s (trial 2) initial v = ______________ m/s Using the energy principle, you shall use your experimental observations to predict the maximum height ∆ymax. Answer the following questions Use m for the mass of the ball, v for the magnitude of the vertical component of the initial velocity, and ∆ymax for the vertical displacement of the ball from the time just after the bounce until the ball reaches its maximum height. Be careful of signs. Do not plug in any numbers until later, though 0’s may be plugged in now. Consider only the ball as the system. SHOW ALL STEPS 1.) Initial kinetic energy (just after bounce): Ki =______________________ Final kinetic energy (at top): Kf =______________________ Force exerted on the ball by the Earth: FbE = < ________, ________, ________ > 2.) Displacement of the ball: Δr = < ________, ________, ________ > 3.) Work done on the ball by the force exerted by the Earth: W =________________ 4.) Write down the energy principle, DO NOT PLUG IN ANYTHING YET. 5.) Using your expressions for the work, displacements, and kinetic energies, solve for ∆ymax in terms of symbols. DO NOT PLUG IN NUMBERS YET. Solve starting from Energy prinCiple. 6.) Now plug in your experimental numbers to predict the maximum heights for your two trials. Show your CalCulations. (trial 1) Predicted ∆ymax = ___________ m (trial 2) Predicted ∆ymax = ___________ m (trial 1) Measured ∆ymax = ___________ m (trial 2) Measured ∆ymax = ___________ m The momentum prinCiple involves time; the energy prinCiple involves distance. Look baCk over your analysis and see how the two principles complement each other, yielding different kinds of information about the phenomenon. If you had Chosen the ball and the earth to be the system would there be any external work done on this system? CHECKPOINT 1: Ask an instructor to check your work for credit. You can read ahead while you’re waiting to be checked off. We are now going to model the bouncing motion of the ball. Save this code with the name: Lab6_BallBounce.py. To get us started, we first need to create our virtual environment within the vPython window. Think about what objects are going to be needed and what shapes we will use to model them. List them below: Great! Now let’s give them some realistic dimensions by assuming we are looking at a 0.5x0.5 m2 area of floor, with a thickness of 0.05 m at a position of <0,-0.025,0>. Measure the diameter and mass of the golf ball you used in the previous portion of this lab and record them below (include units!): D = ____________________ ( ) m = ____________________ ( ) Remember that we have to input a radius into vPython for our golf ball; so don’t forget to divide the diameter by 2. Using the cart lab (Lab 3) as a model, consider the different physical quantities that we need to give the ball and list them below: Now, think of what physical constants we need to add to our program (what’s causing the ball to move, etc.) and write it/them below: Do the same for the simulation parameters needed to model motion. What would be an appropriate delta_t? Ok, now that we have the physical parameters of our model, let’s start thinking how we will use our model in vPython. Answer the following questions using the data from the first part of this lab: Origin of our system: _________________________________ Ball’s initial position: _________________________________ Ball’s initial velocity: _________________________________ Input these into our vPython model. Run it to make sure that you have an appropriate skeleton model. Don’t forget to give the floor some color! Now let’s add some physics! Start the while loop with the appropriate conditional statement; see the previous labs for examples of a good conditional statement. Within the while loop, we will place all of our physics and motion code. Let’s start by getting the ball to move. What force is going to cause the golf ball to fall? In which direction does it act? What is its magnitude? Write the answers to these below and then put them into the appropriate code in vPython. Force ___________________________ Direction ___________________________ Magnitude ___________________________ Inputting into vPython code: F??_hat = vector(??) F??_mag = ?? F?? = ?? Note here that the ?? represents things that you need to fill in, but the code should have this form. What two principles do we need to get the ball to move within the while loop? Write them below. Then, write these in the code, within the while loop, with the appropriate delta_p and delta_r. Remember as you code these into the model that we work with momentum in this lab! Run your model. What happens? Does it match with what happened in our experiment? No, it doesn’t! We need to incorporate the bounce. Let’s start with the simplest possible case: a perfectly elastic bounce. For a perfectly elastic bounce, all of the momentum is conserved. How would this look symbolically, in terms of |pf| and |pi|, the magnitudes of the initial and final momenta? Write it down below: What about the momenta, �! and �!? To incorporate this elastic bounce into the code, we need to add a conditional statement relating the y-coordinate of the ball’s position and the floor. This should be entered within the while loop right before the time is updated. It should look like this: if( ball.pos.y - ball.radius < 0): ball.pos.y = ball.radius ball.p = ?? This code tells the computer that if the ball’s y-coordinate position is less than zero, it will do something to the momentum (to be determined by you!) and sets the balls position back to the top of the floor (why?!). Does the ball actually perfectly bounce off of the floor in our experiment? Can we determine how much energy is lost in the bounce? To find out, we need to analyze the system with a clever choice of initial and final states. We will do so with three different cases. Remember to use symbols, not numbers! CHECKPOINT 2: Ask an instructor to check your work for credit. You can read ahead while you’re waiting to be checked off. Case 1 Let’s start the analysis by looking at the case of choosing the initial state as the beginning of the drop, just after you release the golf ball, and letting the final state be when the ball is just about to strike to floor. System: ____________________________________________ Surroundings: ____________________________________________ Using ONLY symbols, start with the Energy Principle, and solve for the speed of the ball just before it strikes the floor (Hint: This answer should match the one you found in Checkpoint 1 of this lab): What is the numerical value of the speed at this point? v = ________________________________ ( ) What is the momentum at this point? Symbolically and numerically? psymbol = _________________________________________ pnumber = ________________________________ ( ) Case 2 Now, let’s analyze the system with the initial state being at the beginning of the drop, just after you release the golf ball, and let the final state being when the ball comes to a stop at the top of it’s bounce.