New Competition! Rocket LandingThis competition is open to all school students. Send your solution to Greg Bodeker (greg@bodekerscientific.com) by 28 February 2021. Prizes up for grabs:Winner: $600 (plus job interview at Bodeker Scientific if you are interested)Second: $300Third: $100Consider a spacecraft 100 km above
the surface of some planet. It has just been released from the mothership and
needs to descend to the surface carrying two astronauts. The spacecraft weighs
2000 kg and has 1000 kg of fuel onboard. The planet has a gravitational
acceleration of 12 m.s ^{-2} (on Earth it is 9.8 m.s^{-2}).
Burning 1 kg of fuel in one second generates 6000 N of upward thrust.Your job is to write some python code that decides how much fuel to burn, every second, so that the spacecraft lands on the surface under the following constraints: - The upward acceleration must never be more than 3g otherwise the astronauts will pass out.
- The vertical velocity as it touches down must
be greater than 0 m.s
^{-1}and less than 2 m.s^{-1}when the spacecraft lands otherwise you destroy it.
The goal is to land on the surface, under these constraints, with as much fuel in the tank as possible. The planet has an atmosphere. The air is more dense closer to the ground and less dense higher up. This air density will slow your spacecraft because of air friction. As the air becomes more dense so the friction will increase and slow your spacecraft even more. The frictional force depends on the square of the spacecraft velocity. Ah, but if only life were so simple!
This planet has a very turbulent atmosphere with lots of Because the situation is slightly different each time you run the code, we will run your solution 1000 times and will average the fuel left in the tank across all of the runs to get your 'score'. If your algorithm fails more than 50 times out of the 1000 runs you will be disqualified. The person with the highest score will win. Most of the job has already been done for you. Here is some sample code. All you need to do is add code where it is indicated below to generate the fuel to burn (ftb): `import matplotlib.pyplot as plt` `import numpy as np` `thrust_per_kg = 6000` `g = 12.0` `mass_of_fuel = 1000` `mass_of_crew_module = 2000` `time = [0]` `alt = [100000.0]` `actual_speed = [0.0]` `acc = [g]` `fuel_to_burn = [0.0]` `def get_fuel_to_burn(time, alt, speed, fuelmass, density):` ` ftb = 0.0` ` # Students should implement their solution between here`
` ftb = 2.0` ` # and here` ` # The line below prevents more fuel being burnt than is available` ` ftb = ftb if ftb < fuelmass else fuelmass` ` ftb = max(ftb, 0)` ` return ftb` `def generate_atmos_density():` ` """` ` This function generates the atmospheric density profile for the planet. The density profile decays exponentially` ` from the surface to the top of the atmosphere of the planet. The density profile will affect the` ` frictional force that the spacecraft experiences. The profile includes some random waves.` ` :return: an array of 100,001 elements specifying the density from the surface to 100 km altitude every metre.` ` """` ` altitude = np.arange(0,100001,1)` ` dens = 100000 * np.exp(-altitude/7000)` ` # Now generate four 'waves' in the density profile. Each wave will have a randomly generated amplitude, phase and` ` # wavelength within some bounds. These waves are added up and then applied to the density profile` ` waves = np.zeros((100001,), dtype = np.float )` ` for wv in range(1, 5):` ` amplitude = np.random.normal(wv * 0.1, wv * 0.05, 1)` ` # The line below randomly selects the wavelength in metres` ` wavelength = np.random.normal(wv * 4000, wv * 1000, 1)` ` # The line below randomly generate a phase offset` ` phase = wavelength * np.random.random()` ` # Here is the equation that generates the wave` ` wave = amplitude * np.cos(2 * np.pi * (altitude - phase)/wavelength)` ` # Add the four waves` ` waves = waves + wave` ` # If you are in debugging model it can be interesting to plot 'waves' at this point` ` # Now apply the waves` ` dens = dens *(1 + waves)` ` return dens` `#-----------------------------------------------------------------------------------------------------------------------` `# Main program starts here` `#-----------------------------------------------------------------------------------------------------------------------` `dens = generate_atmos_density()` `print('Time, Altitude, Speed, Accel., G-force, Fuel left')` `print('{:4d}, {:8.1f}, {:6.2f}, {:6.2f}, {:6.2f}, {:6.2f}'.format(time[-1],` ` alt[-1], actual_speed[-1], acc[-1], (g - acc[-1]) / 9.8, mass_of_fuel))` `try:` ` while alt[-1] > 0:` ` # Find out how much fuel we will burn in the next second.` ` fuel_to_burn.append(get_fuel_to_burn(time[-1], alt[-1], actual_speed[-1], mass_of_fuel, dens[int(alt[-1])]))` ` # Calculate the deceleration from the fuel burnt in the previous time step` ` acc_from_burn = (thrust_per_kg * fuel_to_burn[-1]) / (mass_of_fuel + mass_of_crew_module)` ` # Calculate the deceleration from friction which depends on the density profile. The frictional force depends on` ` # velocity squared and the density of the atmosphere: acc = force/mass` ` acc_from_friction = actual_speed[-1]**2 * dens[int(alt[-1])] * 5E-7 / (mass_of_fuel + mass_of_crew_module)` ` # Update the mass of fuel` ` mass_of_fuel = mass_of_fuel - fuel_to_burn[-1]` ` # Work out the acceleration at the end of the 1 second period.` ` acc.append(g - acc_from_burn - acc_from_friction)` ` # Work out new speed: v = u + a*t` ` actual_speed.append(actual_speed[-1] + acc[-1])` ` # Work out new altitude: s = ut + 0.5*a*t^2` ` alt.append(alt[-1] - (actual_speed[-2] + 0.5 * acc[-1]))` ` # Update the time` ` time.append(time[-1] + 1)` ` # Print the diagnostics` ` print('{:4d}, {:8.1f}, {:6.2f}, {:6.2f}, {:6.2f}, {:6.2f}'.format(time[-1],` ` alt[-1], actual_speed[-1], acc[-1], (g-acc[-1])/9.8, mass_of_fuel))` ` if (g-acc[-1])/9.8 > 3.0:` ` raise Exception('G-force exceeded 3.0.')` ` landing_speed = actual_speed[-1]- (actual_speed[-2] - actual_speed[-1])/ (alt[-2] - alt[-1]) * alt[-1]` ` if (landing_speed < 0.0) or (landing_speed > 2.0):` ` raise Exception('speed on landing was too high.')` ` print('Well done! you landed successfully.')` ` print('Landing speed was {:6.4f} m.s^-1'.format(landing_speed))` ` print('Final fuel left in tank={:6.3f} kg'.format(mass_of_fuel))` `except Exception as err:` ` print('Sorry your solution failed because {}'.format(err))` `# Make plot of altitude and speed` `fig, ax = plt.subplots()` `ax.plot(time, alt, color='red')` `ax.set_xlabel('Time (seconds)', fontsize=12)` `ax.set_ylabel('Altitude (m)', color='red', fontsize=12)` `ax2=ax.twinx()` `ax2.plot(time, actual_speed, color='blue')` `ax2.set_ylabel('Speed (m/s)', color='blue', fontsize=12)` `plt.show()` `plt.close()` `# Make plot of acceleration and fuel burnt` `fig, ax = plt.subplots()` `ax.plot(time, acc, color='red')` `ax.set_xlabel('Time (seconds)', fontsize=12)` `ax.set_ylabel('Acceleration (m/s^2)', color='red', fontsize=12)` `ax2=ax.twinx()` `ax2.plot(time, fuel_to_burn, color='blue')` `ax2.set_ylabel('Fuel burnt (kg)', color='blue', fontsize=12)` `plt.show()` `plt.close()` `#-----------------------------------------------------------------------------------------------------------------------` `# Main program ends here` `#-----------------------------------------------------------------------------------------------------------------------` Clearly what is written above is not a successful algorithm - the spacecraft hits the surface at well over 2 m.s ^{-1} with the result that the astronauts need to be scraped off the floor with a spatula. Your job is to modify the algorithm between the lines indicated in the code above so that the landing occurs within the constraints detailed above. The winner will be the person whose algorithm lands with the most fuel left in the tank, within the constraints. |

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