Exercise 7: Thrust and the Rocket Equation
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1 Astronomy 102 Name: Exercise 7: Thrust and the Rocket Equation Parts of this exercise were taken from NASA s Beginner s Guide to Rockets: Rocket Propulsion Activity, edited by Tom Benson. Objective: To develop mathematical relationships between mass, thrust and distance traveled by a reaction motor. Needed: Round balloon, short length of drinking straw, string and tape Procedure: 1. With a piece of string cut to fit around the balloon, blow up the balloon completely and measure the circumference of the balloon. Record it in the table on the next page. Keep the balloon inflated by holding it (not tying it) closed. 2. Set aside the string for future use. 3. Slide the straw onto a fairly long (at least four meter) piece of string. 4. Choose two people from your team to hold each end of the string taut and level. 5. Place two open loops of tape onto the straw. 6. Attach the balloon to the two pieces of tape on the straw as shown. The neck of the balloon should be parallel to the string. 7. Release the neck to let the air rush out of the balloon. 8. In the table shown below, record (a) the distance that the balloon traveled (as marked on the string) and (b) the length of time the balloon was moving. 9. Repeat the process, filling the balloon with different amounts of air (the table shows suggested fill s, but you can estimate your own). Use the piece of string you used to measure in the first step. 10. Calculate the radius and volume of the balloon with the different circumferences using the equations below (note that we are pretending the balloon is spherical, even when it is not): radius =!"#!$%&'#'(!'!! volume =!! π r!
2 11. Calculate the mass of air that was originally in the balloon by using the density of air at sea level and 15 C (this value I got from the Wikipedia article): mass = (density) (volume) = ( g/cm 3 ) (volume) fill circumference radius volume (cm 3 ) Distance travelled Flight time (s) Mass of air in balloon (g) ¾ ½ ¼ On a piece of graph paper, plot the air volume in the balloon (x-axis) versus the distance the balloon travelled (y-axis). Remember to choose sensible spacings and tick marks to fill the graph paper, and to label the axes with units. What mathematical relationship, if any, connects the volume and the distance? If appropriate, write a mathematical expression connecting volume and distance. The goal for this lab is to determine the thrust of the balloon. One of the characteristic measures of any rocket s performance is its thrust, which is also called the reaction force. Like force, the units of thrust is Newtons (N). Thrust is calculated as the product of the exhaust velocity of the gases coming out of the nozzle and the rate at which mass is expelled out of the nozzle. Mathematically: Average thrust (N) = exhaust velocity (m/s) (!!!"#$!"!"##!" ) (!!!"#$!"!"#$! )
3 Symbolically, the equation looks like: Taverage = ve!! To get the exhaust velocity, we ll have to use the rocket equation from last week:!! Δv = v! ln!!"!#!$%!!"#$% or v! =!!"#$%!!"!#!$%!"!!"!#!$%!!"#$% What should mfinal be, for our experiment? Measure mfinal and record it (in grams). How will minitial be calculated? The velocities on the right side of the second equation refer to the rocket (balloon) itself. How will the average velocity of the balloon (call it vfinal) be calculated? Hint: you will need two of the quantities from the table on the previous page. What is vinitial? This simplifies the second equation above a bit. To be able to use the second equation, we need to convert the units of the velocities to m/s from cm/s. How can this be done easily? Hint: How many meters in one centimeter? We also need to convert the units of masses into kg from g. How is this done?
4 From the information in the previous table, calculate the thrust by filling in the table below: fill vfinal (m/s) minitial (kg) mfinal (kg) ve (m/s) ¾ ½ ¼ Now calculate the thrust: fill Change in mass of balloon (kg) Flight time (change in time) (s) Rate of change in mass (kg/s) Average thrust (N) ¾ ½ ¼ For an air-propelled engine, the Boeing ER s GE90-115BL engine yields 512,000 N of thrust, according to Boeing. How did the balloon compare?
5 What is the optimal balloon fill, in order to achieve maximum thrust? What factors did we not take into account that would have influenced the calculation of thrust? Pick one and explain how that affected the thrust (i.e., follow a line of reasoning that leads to a conclusion about whether the thrust calculated was too high or too low).
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