A multi-stage atmospheric re-entry vehicle is being designed as part of a mission to land a robotic rover to the surface of Mars. As the re-entrjy vehicle approaches the upper atmosphere of Mars, a heat shield is used to slow the vehicle down via drag (first stage). When the re-entry vehicle reaches an altitude specified by hparachute, a parachute system is deployed total drag coefficient on the re-entry vehicle, reducing its speed to a steady terminal velocity (second stage) When the vehicle reaches an altitude of 500m above the ground, a rocket propulsion system is activated and used to reduce the velocity of the vehicle to zero at the point of touchdown at the surface. which further increases the parachute rocket (500m) Assuming a vertical trajectory, an equation of motion can be developed for the first and second stages (heat shield and parachute) by considering the balance of weight forces and drag forces acting on the vehicle such that: where h is the vehicle's altitude above the ground, v is the downwards velocity, gmars is the surface gravitational acceleration on Mars (3.69 m/s*), is the Martian atmospheric density (kg/m3) and kirag is a constant, dimensioned drag coefficient. Before the parachute is deployed, kirag - 0.0021 m2/kg (heat shield) and after the parachute is deployed kdrag = 0.0295 m2/kg. The atmospheric density of Mars varies with altitude and can be approximated by: T- -31.0 0.000998h P 0.699e-0.00009h 0.1921(T 273.1) where T is atmospheric temperature (oC), P is atmospheric pressure (KPa), h is altitude (m) and p is the Martian atmospheric density (kg/m3). As an engineer on the design team, you are asked to perform calculations to determine at what altitude hparachute should the parachute be deployed for the second stage. Develop a MATLAB function that uses an Euler numerical integration method to solve the equations of motion shown above for the first and second stages of re-entry. Your function should take as an input the initial velocity and altitude of the vehicle in the upper atmosphere and the altitude hparachute at which the parachute should be deployed (hence changing the drag coefficient of the vehicle). Your function should output three arrays t, h and v which represent the altitude and velocity profiles vs. time of the re-entry vehicle from the initial time up until the point at which the vehicle is at an altitude of h- 500m above the ground, when the third stage rocket begins (you do not need to model the third stage of the system). Use a timestep of dt-0.05s. Inside your function you should iteratively evaluate the velocity and altitude at each timestep by integrating the equation of motion shown above (15 Marks) i. ii. Use your developed function in (i) to produce plots of a. The velocity of the vehicle vs. time b. The velocity of the vehicle vs. altitude for three proposed values of hparachute at 20km, 12km and 4km. Plot the three trajectory cases in the one figure (with different colours/line markers) for each of (a) (first figure) and (b) (second figure) and include a legend in each plot. Use initial values for the starting altitude at ho- 40km and starting velocity vo 1850 m/s (downwards). (5 Marks) Perform calculations for the trajectory as in (i) but where the Martian atmospheric density is approximated as being constant. You may either create a second MATLAB function that performs the same calculations as in (i) but instead approximates the density as constant (set to the value at the surface where h - 0), or you may create an extra input to your existing function that acts as a "flag" as to whether calculations should proceed with the variable density modelling (as per the equations above) or with a constant density. Use a loop around developed function/s to compute the final vehicle velocity at h -500m for a range of values of hparachute from 2km to 20km. Show two different curves in the one figure one considering the variable density model and one considering the approximation of the density as constant. (7 Marks) iii. In no more than half a page, discuss the effect of drag and atmospheric density on the re-entry vehicle trajectory and approximate the minimum altitude at which the parachute could be deployed such that the vehicle velocity at h- 500m (end of the second stage) does not exceed 150m/s, using you MATLAB plots. (8 Marks) iv. A multi-stage atmospheric re-entry vehicle is being designed as part of a mission to land a robotic rover to the surface of Mars. As the re-entrjy vehicle approaches the upper atmosphere of Mars, a heat shield is used to slow the vehicle down via drag (first stage). When the re-entry vehicle reaches an altitude specified by hparachute, a parachute system is deployed total drag coefficient on the re-entry vehicle, reducing its speed to a steady terminal velocity (second stage) When the vehicle reaches an altitude of 500m above the ground, a rocket propulsion system is activated and used to reduce the velocity of the vehicle to zero at the point of touchdown at the surface. which further increases the parachute rocket (500m) Assuming a vertical trajectory, an equation of motion can be developed for the first and second stages (heat shield and parachute) by considering the balance of weight forces and drag forces acting on the vehicle such that: where h is the vehicle's altitude above the ground, v is the downwards velocity, gmars is the surface gravitational acceleration on Mars (3.69 m/s*), is the Martian atmospheric density (kg/m3) and kirag is a constant, dimensioned drag coefficient. Before the parachute is deployed, kirag - 0.0021 m2/kg (heat shield) and after the parachute is deployed kdrag = 0.0295 m2/kg. The atmospheric density of Mars varies with altitude and can be approximated by: T- -31.0 0.000998h P 0.699e-0.00009h 0.1921(T 273.1) where T is atmospheric temperature (oC), P is atmospheric pressure (KPa), h is altitude (m) and p is the Martian atmospheric density (kg/m3). As an engineer on the design team, you are asked to perform calculations to determine at what altitude hparachute should the parachute be deployed for the second stage. Develop a MATLAB function that uses an Euler numerical integration method to solve the equations of motion shown above for the first and second stages of re-entry. Your function should take as an input the initial velocity and altitude of the vehicle in the upper atmosphere and the altitude hparachute at which the parachute should be deployed (hence changing the drag coefficient of the vehicle). Your function should output three arrays t, h and v which represent the altitude and velocity profiles vs. time of the re-entry vehicle from the initial time up until the point at which the vehicle is at an altitude of h- 500m above the ground, when the third stage rocket begins (you do not need to model the third stage of the system). Use a timestep of dt-0.05s. Inside your function you should iteratively evaluate the velocity and altitude at each timestep by integrating the equation of motion shown above (15 Marks) i. ii. Use your developed function in (i) to produce plots of a. The velocity of the vehicle vs. time b. The velocity of the vehicle vs. altitude for three proposed values of hparachute at 20km, 12km and 4km. Plot the three trajectory cases in the one figure (with different colours/line markers) for each of (a) (first figure) and (b) (second figure) and include a legend in each plot. Use initial values for the starting altitude at ho- 40km and starting velocity vo 1850 m/s (downwards). (5 Marks) Perform calculations for the trajectory as in (i) but where the Martian atmospheric density is approximated as being constant. You may either create a second MATLAB function that performs the same calculations as in (i) but instead approximates the density as constant (set to the value at the surface where h - 0), or you may create an extra input to your existing function that acts as a "flag" as to whether calculations should proceed with the variable density modelling (as per the equations above) or with a constant density. Use a loop around developed function/s to compute the final vehicle velocity at h -500m for a range of values of hparachute from 2km to 20km. Show two different curves in the one figure one considering the variable density model and one considering the approximation of the density as constant. (7 Marks) iii. In no more than half a page, discuss the effect of drag and atmospheric density on the re-entry vehicle trajectory and approximate the minimum altitude at which the parachute could be deployed such that the vehicle velocity at h- 500m (end of the second stage) does not exceed 150m/s, using you MATLAB plots. (8 Marks) iv