Modeling the Motion of a Projectile in Air

Transcription

1 In this lab, you will do the following: Modeling the Motion of a Projectile in Air analyze the motion of an object fired from a cannon using two different fundamental physics principles: the momentum principle and the energy principle; model the motion with a computer program to simulate this motion, both with and without air resistance, to see the effects of a dissipative force on motion, and to decompose the forces into parallel and perpendicular components; and plot the energy graphs for this system to see the effects of dissipative forces on energy. 1 Preparation: Analysis A ball (mass 50 kg) is fired from a cannon with an initial velocity of 60,30,0 m/s at a location of 500,100,0 m. For simplicity, let s first consider the force of air resistance on the ball to be negligible. Answer the following questions on a whiteboard. Start from fundamental physics principles and be clear about your choice of system. Do not use any pre-derived formulas. Be prepared to show all steps in your work to your TA, not just the final answer. 1. Starting from the energy principle, find the speed (magnitude of velocity) of the ball just before it hits the ground (defined by the plane y=0). 2. Starting from the momentum principle, find the x-component of the velocity of the ball just before it hits the ground. (Hint: You are near the surface of the Earth) 3. Starting from the momentum principle and the position update formula, find the amount of time it takes for the ball to hit the ground after it is fired from the cannon. 4. How far has the package traveled horizontally during the time it was falling? Draw a diagram showing the path of the ball during this time. 5. Imagine that the mass of the ball was 100 kg instead of 50 kg (all else being the same). Qualitatively, how would your answers to parts 1 through 4 change? Explain briefly. 6. Finally, let s imagine the force of air resistance (see page 245 of the textbook) to be substantial enough that it cannot be ignored (which is closer to reality in a case such as this.) In your own words, how would your answers to parts 1 through 5 change? Would the mass of the ball affect your answers in this case? Draw a new diagram showing the path ball for the case of nonzero air resistance. 1

2 Write your answers to the above questions on a sheet of paper, so that you can compare them to the results you get using the computer model. Compare your work with a neighboring group. Have an instructor check your work before proceeding to the program. 2 Modeling the motion without air resistance Download the VPython script from the WebAssign assignment. Read through the code carefully before moving on. Make sure you understand the structure of the code, and meanings of the various parameters. 2.1 Writing the program First, we will use the program to model the motion of the fired ball in the case of no air resistance. You will need to insert the proper code into the while loop of the program. Your code should do the following: 1. calculate the net force on the ball 2. update the ball s momentum 3. update the position of the ball 4. draw a trail for the ball 5. determine the parallel component of the net force 6. determine the perpendicular component of the net force 7. update the locations and axes of the arrows representing these components 8. update time After the loop, add code to print the final values of: the time the balls speed the x- and y-component of the ball s velocity the x-component of the ball s position When finished, run the program. 2

3 2.2 Questions about the model Answer these questions on your whiteboard. 1. Observe the trajectory of the ball. What do you notice? What is the ball s final position? Does this agree with what you predicted in your written analysis? 2. Compare your calculated final speed and total time of the drop to those predicted by the computer model. 3. Change the mass of the ball to 100 kg. Does this make any difference to the motion? 4. Increase the time step by a factor of 10 of the original value, run the program. Discuss the resulting 5. Increase the time step by a factor of 100 of the original value, run the program. Discuss the resulting 6. Return the time step to its original value. 3 Modeling the motion with air resistance Now modify the computer program to include the force of air resistance on the falling package. An approximate formula for the force of air resistance on an object is given by: F air = 1 2 CρAv2ˆv where C is the drag coefficient, ρ is the density of air (called D in the program), A is the cross sectional area of the object, and v is the object s speed. Note that the direction of the force is opposite to object s velocity. The drag coefficient depends on the shape and surface properties of the object. We will assume our ball is a smooth sphere with a drag coefficient of about 0.1. The ball has a radius of 1 m, and the density of air at sea level is about 1.3 kg/m 3. Check the Constants section of your program to make sure these values are entered correctly. Once you have modified the program, run it, and answer the following questions on your whiteboard. 1. Observe the trajectory of the ball. What do you notice? What has changed from the case without air resistance? Compare the final position of this ball with the one from the previous section. Does this agree with what you predicted in your written analysis? 2. How has the final velocity before impact changed? Does this make sense? 3. Try using a mass of 100 kg and 200 kg for the ball. Does changing the mass make a difference to the motion? Why? 4. Try different values of the drag coefficient between 0.05 and 1.0 to see what effect it has on the Compare your work with a neighboring group. Have an instructor check your work before proceeding to the next section. 3

4 4 Plotting energy graphs We will now add code to plot graphs of kinetic energy, potential energy, and their sum for the system of the ball and Earth (but not the air). Before the loop, type in the following statements: Kgraph=gcurve(color=color.red) Ugraph=gcurve(color=color.cyan) KplusUgraph=gcurve(color=color.yellow) Now in the loop of the program, after updating the position of the objects, but before updating time, add code to calculate the kinetic energy of the ball, K, and the gravitational potential energy, U. Note that you can use the approximation for gravitational potential energy near the Earth s surface, mgy. In fact, this formula gives the difference in potential energy from a fixed reference point (i.e., the ground), not the actual potential energy. This is fine because we are only interested in how the potential energy changes in this case. Immediately after the code that calculates K and U, add statements to plot these values: Kgraph.plot(pos=(t,K)) Ugraph.plot(pos=(t,U)) KplusUgraph.plot(pos=(t,K+U)) 5 Numerical Experiments Let s first examine the case of zero air resistance. Do not erase the air resistance statements in your program you can temporarily set the value of the drag coefficient to zero. Run the program. Set C to 0 to remove the effects of and run the program. 1. Do the graphs of K and U make sense? 2. Is the graph of K + U for the Earth-ball system what you expect? Now reset C to 0.1 to include air resistance again and run the program. 1. What has happened to the graph of K + U? Where is the energy going? 2. How has the graph of K changed from the no air resistance case? 3. Notice the rate of change of K + U (the slope of this curve) is highest when the kinetic energy is highest. If this is hard to notice, try changing the drag coefficient from 0.1 to 0.4. Why is this the case? (Think about which quantities the air resistance force depends on.) 4. Discuss the phenomenon of energy dissipation as it pertains to this numerical experiment. Use specific examples (e.g., C=0 vs C=0.05). 4

5 6 Turn in your program to WebAssign You will need this program for a homework assignment. Before uploading it, make sure it works properly. Make sure everyone in the group agrees that the program is correct. Check with a neighboring group and your TA. Everyone should turn in the program to WebAssign, and make sure everyone has a copy of the program. You will need this program for a homework assignment. When you turn in a program to WebAssign, be sure to follow the instructions given there, which may sometimes ask you to change some of the parameters in your program. You can retrieve the program from WebAssign once you have uploaded it. Have an instructor check your work for credit. Submit your program to WebAssign and answer any WebAssign questions for this lab. Last edited: Friday 27 th August, 2010 at 1:33pm 5