###
Question Description

### Lecture 2, Rocket Science

Some physical problems are easy to solve numerically using just the basic equations of physics. Other problems may be very difficult. Consider a specific model rocket with a specific engine. Given all the data we can find, compute the maximum altitude the rocket can obtain. Yes, this is rocket science. Most physics computation is performed with metric units. units and equations Estes Alpha III Length 12.25 inches = 0.311 meters Diameter 0.95 inches = 0.0241 meters Body area 0.785 square inches = 0.506E-3 square meters cross section Cd of body 0.45 dimensionless Fins area 7.69 square inches = 0.00496 square meters total for 3 fins Cd of fins 0.01 dimensionless Weight/mass 1.2 ounce = 0.0340 kilogram without engine Engine 0.85 ounce = 0.0242 kilogram initial engine mass Engine 0.33 ounce = 0.0094 kilogram final engine mass Thrust curve Total impulse 8.82 newton seconds (area under curve) Peak thrust 14.09 newton Average thrust 4.74 newton Burn time 1.86 second Initial conditions: t = 0 time s = 0 height v = 0 velocity a = 0 acceleration F = 0 total force not including gravity m = 0.0340 + 0.0242 mass i = 1 start with some thrust Basic physics: Fd = Cd*Rho*A*v^2 /2 two equations, body and fins Fd is force of drag in newtons in opposite direction of velocity Cd is coefficient of drag, dimensionless (depends on shape) Rho is density of air, use 1.293 kilograms per meter cubed A is total surface area in square meters v is velocity in meters per second (v^2 is velocity squared) Fg = m*g Fg is force of gravity toward center of Earth m is mass in kilograms g is acceleration due to gravity, 9.80665 meters per second squared Ft = value from thrust curve array at this time, you enter this data. index i, test i>18 and set Ft = 0.0 Do not copy! This is part of modeling and simulation. start with first non zero thrust. F = Ft - (Fd body + Fd fins + Fg) resolve forces a = F/m a is acceleration we will compute from knowing F, total force in newtons and m is mass in kilograms of body plus engine mass that changes dv = a*dt dv is velocity change in meters per second in time dt a is acceleration in meters per second squared dt is delta time in seconds v = v+dv v is new velocity after the dt time step (v is positive upward, stop when v goes negative) v+ is previous velocity prior to the dt time step dv is velocity change in meters per second in time dt ds = v*dt ds is distance in meters moved in time dt v is velocity in meters per second dt is delta time in seconds s = s+ds s is new position after the dt time step s+ is previous position prior to the dt time step ds is distance in meters moved in time dt m = m -0.0001644*Ft apply each time step t = t + dt time advances i = i + 1 print t, s, v, a, m if v < 0 quit, else loop Ft is zero at and beyond 1.9 seconds, rocket speed decreases Homework Problem 1: Write a small program to compute the maximum height when the rocket is fired straight up. Assume no wind. In order to get reasonable consistency of answers, use dt = 0.1 second Every student will have a different answer. Some where near 350 meters that is printed on the box. +/- 30% Any two answers that are the same, get a zero. Suggestion: Check the values you get from the thrust curve by simple summation. Using zero thrust at t=0 and t=1.9 seconds, sampling at 0.1 second intervals, you should get a sum of about 90 . Adjust values to make it this value in order to get reasonable consistency of answers. The mass changes as the engine burns fuel and expels mass at high velocity. Assume the engine mass decreases from 0.0242 kilograms to 0.0094 grams proportional to thrust. Thus the engine mass is decreased each 0.1 second by the thrust value at that time times (0.0242-0.0094)/90.0 = 0.0001644 . mass=mass-0.0001644*thrust at this time. "thrust" = 0.0 at time t=0.0 seconds "thrust" = 6.0 at time t=0.1 seconds. "thrust" = 0.0 at and after 1.9 seconds. Important, rocket is still climbing. Check that the mass is correct at the end of the flight. 0.0340+0.0094 Published data estimates a height of 1100 feet, 335 meters to 1150 feet, 350 meters. Your height will vary. Your homework is to write a program that prints every 0.1 seconds: the time in seconds height in meters velocity in meters per second acceleration in meters per second squared force in newtons mass in kilograms (just numbers, all on one line) and stop when the maximum height is reached. Think about what you know. It should become clear that at each time step you compute the body mass + engine mass, the three forces combined into Ft-Fd_body-Fd_fins-Fg, the acceleration, the velocity and finally the height. Obviously stop without printing if the velocity goes negative (the rocket is coming down). The program has performed numerical double integration. You might ask "How accurate is the computation?" Well, the data in the problem statement is plus or minus 5%. We will see later that the computation contributed less error. A small breeze would deflect the rocket from vertical and easily cause a 30% error. We should say that: "the program computed the approximate maximum height." Additional cases you may wish to explore. What is the approximate maximum height without any drag, set Rho to 0.0 for a vacuum. What is the approximate maximum height using dt = 0.05 seconds. What is the approximate maximum height if launched at 45 degrees rather than vertical, resolve forces in horizontal and vertical