- Rocket Dynamics Objective Use Excel To Calculate The Position Velocity And Acceleration Of A Vertically Launched Rock 1 (48.65 KiB) Viewed 21 times
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- Rocket Dynamics Objective Use Excel To Calculate The Position Velocity And Acceleration Of A Vertically Launched Rock 7 (69.42 KiB) Viewed 21 times
Fgravity is described by Newton's law, Fgravity -GM m (Re+y)2, where G is the gravitational constant, Me is the mass of the earth, m is the mass of the rocket, and Y is the altitude. Note that Re ty is used because this is the total distance from the center of the carth to the rocket when the rocket is at an altitude y, while the minus sign in front is because the direction is down. Fair is the air friction force and is described by the law ,P = -4.5DAp)v2 where D is an aerodynamic drag constant, A is the rocket's cross-sectional area, v is the speed, and p is the atmospheric density. The atmospheric density decreases with altitude approximately according to the equation. p - Poe-ky, where po is the density at sea level and k is an atmospheric constant. The minus sign is used because air friction is downward as the rocket moves upward. The air friction force now becomes, Fair=-(5DApoc-ky)*v2. Entering in all the force terms and solving for, a, the acceleration: a=dv/dt (Vrelºdm/dt)(m-GM (Re Y)2-(.5DApoc-ky-2/m. (2) Note that the first term is the acceleration due to the thrust, the second is the acceleration due to gravity, and the third is the acceleration due to air friction. This second-order differential equation is impossible to solve analytically since so many things are changing (m, y, V, and a), and the acceleration and the velocity are derivatives of y. It can only be solved exactly in some very special cases such as zero friction and constant gravity. However, there is a way to obtain an approximate but very accurate description of the motion (even in the general case for y(t) and v(t)), by using a simple numerical method which can be programmed into a spreadsheet. The basic concept of the method is to consider what happens during very small time intervals, At. The acceleration during these small time intervals will be approximately constant, and we can use the simple equations of constant acceleration. The accuracy of the method will be dependent upon how small we make St. For this problem, the method works as follows: Atto мо initial mass of rocket
Yo 0 20 = (vreledm dt)mo - GM/R ? from equation (2) YOSO Note that at air is zero initially since V, is zero. At later mi-mo-dm dt)*At Y1 yotv,At+ 5a At2 a = (Vrele dm/dt) m; - GM/(Re+ y)2 = (-5DAP.e-ky vo +At)2 mi Vivo +.5(aga, At Note that in aj, the velocity in the air friction term is calculated using the equation v=vaAt. A better approximation for v, once a2 has been calculated, is obtained in vi using 5(ata) which is the average acceleration in the time interval. Similarly, the mass position, acceleration, and velocity can be found at later times by continuing this iterative process. TABLE 1: An example of populating the excel table t m Y V o initial mass of rocket YO V=0 Vreldm 20- dt GM mo 2. Re واليا ** *و* dm mo dt Vreldm solar V,V, Sagt At, 5[+]A دود ۵۰. الا 1 OM m1 [Retrof 5 do A mi Vreld dryt. 22" m2 [Retrat 5odo A. m2 Atmzm- OM V2V, 5[*] drn At Y-Y, VA 5a4 درد ۷۱۰ و 2 * Study the table 1 and explain it to your partner. Excel Layout & Procedure
1) Click and open the Rocket Dynamics Spreadsheet. 2) In column A. you can enter and change the rocket's initial mass (mo), the rate of mass loss (dmdt), the exhaust velocity (rel), the time interval (At), the mass of the earth (Me). the gravitational constant (G). the air friction shape constant (c), and the cross-sectional area of the rocket (A). Column B is the mass column. It starts with Mo and each successive row subtracts (dm dt) At 3) Column C starts with the initial time and adds At to each successive row. 4) Column D, the vertical position, starts with y=0 and then uses the equation for constant acceleration in subsequent rows. 5) Duplicate columns for time (E,G. 1. k) are included to facilitate plotting. 6) Column F, H. I calculate a air a gravity, a thrust using equation 2. Three separate columns are used so that you can study the relative sizes of these terms. 7) Column Ladds up the accelerations including minus signs for the negative accelerations of gravity and air friction 8) Column N calculates velocities starting from zero using an equation of the form: v2=v1+Sajta2)at. 9) Click to each column, study the equations and how they are arranged on the spreadsheet. 10) Enter the following initial conditions and constants into column A of the spreadsheet: Mo-25,000 kg dm dt= 100 kg sec Vrel 4000 m/sec Me=5.98E24 G-6.67E-11, 63, At = 2.0s A1 11) Study cach of the columns to see how rapidly, and in what way, cach of the variables is changing Q1. Find the mass at 2 seconds. Does this value make sense? Q2. Why is the initial air friction zero? Q3. Find the total acceleration, velocity, and position, at 0, 50, 100, and 150, and 200 seconds. How much distance, velocity, and acceleration did the rocket gain in the first 10 seconds? The last 10 seconds? Make a record of the results.
12) Record the absolute values of the accelerations due to thrust, gravity, air friction and atotal at t= 0.100, and 200 seconds. Q4. About how big is air friction at 200 seconds? QS. PLOT ALL GRAPHS IN THE REST OF THIS SIMULATION USING XY SCATTER TYPE THAT HAS DATA POINTS CONNECTED BY SMOOTH LINES WITHOUT MARKERS. 13) Select the data in columns E and F. and plot dair vs. t (Don't forget to title it). Enlarge and position the plot until it fills the entire screen except column A. Q6. Explain the shape. Q7. When does it peak out? Q8. At what speed and altitude? Q9. What is the maximum value of air? Q10). How does this compare to g (9.8 ms)? What % of g is it? Make a printout of the graph and column A (only the first page of your spreadsheet) for your lab report, then delete the graph. 14) Select the data in columns 1 and J. and plot a thrust vs. t. Enlarge and position the plot until it fills the entire screen except column A. Q11. Explain the shape Q12. Determine the thrust accelerations at 10, 50, and 100 seconds. Make a printout of just the graph and first page for your lab, then delete the graph. 15) Select the data in columns K and L, and plot & title atotal vs. t Enlarge and position the plot until it fills the entire screen except column A. Make a printout of just the graph and first page for your lab report. Now a. Change Vrel on the spreadsheet and note the change in the curve. b. Do the same with dm/dt. Don't change it so much that you run out of mass! c. Now change c and A by small amounts. How do these changes affect the rocket? d. Delete just the graph. e. Change the constants back to the original values in step 10. 16) Select the data in columns C and D, plot & title y vs. t, and comment on the shape. Make a one-page printout for your lab report, then delete just the graph.
17) Select columns M and N. plot v vs. t, and comment on the shape. a. Change whatever constants you like and study the effects on the curve. 6. Make a one-page printout for your lab report Q13. Explain what you have learned. c. Delete just the graph. 18 Set the constants to the original values in step 10. Q14. Find the velocity at 200 seconds Eliminate air friction by making c = 0. Q15. Again find the velocity at 200 seconds. Q16. By what % did the velocity change by ignoring air friction? Q17. How big a factor was air friction in determining speed? Eliminate gravity by making G=0, keep e = 0, Q18. What is the velocity at 200 seconds now? Q19. How big a factor was gravity? 19) With the same settings as the end of step 18 (c=0, G=0, At-2 sec.), find y and V at 30 seconds. a. Do the same for At=1 second. b. Repeat for At = 5 seconds. c. Compare these three results for V to the theoretical exact answer for no gravity and no air friction (v=rel"ln(m, mg), where m, is the initial mass and mfis the mass)at 50 seconds. Q20. Find the difference between the answers from the approximate numerical Excel method and that of the exact result Q21. What do you conclude? 20) Using the same settings as in step 10 (including nonzero G and nonzero s), reprogram the spreadsheet so that due to gravity is a constant. You don't have to make 100 changes. Make the appropriate changes only in the column for a due to gravity in the following way a. Click cell H2, and make it a constant (simply type in 9.79). b. Hit the return key. This enters the equation into 2. c. Copy this constant into the other cells in column H: d. Click H2 c. Go to the last data cell in column H and place the pointer inside the box. Do not click!
f. Press and hold down the shift key, then click the cell. This should highlight the column in black 2. Go to the Edit menu and fill down. This should cause the first equation to "fill down into the other cells of the column. If it doesn't start over Q22. Find y at 200 seconds and compare to the original value with gravity changing. (from step11) Q23. Find the % difference. Q24. How good an approximation is it to assume gravity is constant in the rocket problem? 21) Reprogram Excel so that the air density stays constant at 1.53 kg/m (the value at sea level). Keeping gravity constant as in step 20. [Remember that the program assumed p-poe ky and Fair - SDApoc-ky-2] a. Modify one cell and use the Fill down method. Q25. Find y at 200 seconds and compare to the value in step 20 Q26. Find the % difference. Q27. How good an approximation is it to assume that atmospheric density is constant? 22) Quit the program but do not save any changes. This will return it to the way it was. 23) Find space shuttles maximum speed and determine how long it would take to get to the nearest star Alpha Centauri at that speed. 24) Some rocket scientists are calling for nuclear fission rockets for a Mars Mission. What are their advantages and disadvantages? What's wrong with a chemical rocket for a Mars mission? 25) Google nuclear fusion rockets and discover what the future may hold for trips to nearby stars. What kind of velocities are they predicting? Explain theoretically how they could work.