("MUST KNOW": basic first order ODE)

TASK-1

("MUST KNOW": basic first order ODE)

Water is leaking out of a reservoir so that the height of the water, h (in metres), satisfies the differential equation:dh/dt = - k h^(1/2),where time, t, is measured in hours and k is a particular case constant.Determine time (in hours), required for the height of the water to drop to 0.2 m, solving differential equation of the system numerically, using MATLAB.Use the following particular case data: initial water level was h0 = 3.5 m, and after 4 hrs it dropped to 3 m.

119062531940500Figure-1: Leaking water reservoir.

TASK-2

(1st Order Ordinary Differential Equations, ode45, MATLAB)

Consider four ponds connected by streams. Initially, the ponds were pristine.At time t=0, after environmental accident, the second and third ponds are getting polluted with the sources, releasing pollutant substance with the following rates: f2=1 kg/h and f3=4 kg/h. Pollution spreads via the connecting streams to the other ponds, as shown in the Figure. For this task, formulate associated differential equations in terms of x1(t), x2(t), x3(t) and x4(t) - time functions, describing pollutant amounts in each lake. Solve problem numerically, using ode45 MATLAB procedure, and DETERMINE the amount of pollutant (in kg) in lake #2 towards the end of 48 hours of contamination. For a specific numerical example, take f12/V1=f41/V4=0.05; f23/V2=f34/V3=0.02; f24/V2=0.03 [all in 1/h].

91440060960000Figure-2: Four ponds #1, #2, #3 and #4 of volumes V1, V2, V3, V4 [all in m^3], connected by streams with specific flow rates f12, f13, f23, f34, f41 [m^3/h].

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TASK-3

(Modelling, FEM, matrices, MATLAB loops, eig)

Consider a multi-degree-of-freedom undamped mass-spring system shown. Assume, that the following parameters of the system are known and given: N=40; m=12 [kg]; k=5,000 [N/m]. FIND numerical value of the 7th natural frequency omega_7 of the system [in rad/s]. Note: in this system, in addition to the consecutive masses connection via springs, there are additional connections between masses with adjacent even numbers.

Figure-3: Multi-Degrees-of-Freedom Mass-Spring undamped System

0-317500

PLEASE, READ ADDITIONAL RECOMMENDATIONS AND ENTER YOUR ANSWER for TASK-3 BELOW:*Round your result up to the four digits after the dot and select one of the most closely matching (within plus/minus 5%) numerical answer.Note: Your correct answer may be discounted if you do not submit a properly working MATLAB script, producing correct answer.

23.0210 rad/s

8.8020 rad/s

83.2301 rad/s

1.0056 rad/s

None of the above

I DO NOT KNOW how to solve this task

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TASK-4

(MATLAB: "Integrated knowledge", including ODE, loops, arrays, plots, etc.)

Drone "A" is flying with constant speed "U" along the straigh line, characterised with constant distance "H" from this surface.Angle of inclination of the surface "theta" is known.Orthogonal projection of the drone "A" on the surface is denoted as "B" (i.e. AB=H).When drone is discovered by the drone distroyer "P", the drone is locatated uphill at the distance "D" from the point "B" (i.e. PD=D) and this time is assumed to be zero. Some time later, at the proper time, projectile is launched towards the drone at angle "beta" relative to the inclined ground surface, as shown in the Figure. Determine time [in seconds] from the discovery of drone to its intercept at Point-1, as per the Figure.Assume NO air resistance.Also, assume the following parameters: velocity of the drone U=44 [m/s] initial velocity of the projectile v0=100 [m/s] inclination angle of the ground surface theta=15 [deg] projectile shooting angle in the inclined coordinates beta=55 [deg] hight level of the trajectory of the drone AB=H=200 [m] distance from projectile launcher "P" to the point "B" (being orthogonal projection of the drone "A" on the ground surface) PB=D=1,800 [m] intercept Point: number 1 (i.e. point with smaller projectile flight time to intercept)Recommendation: you may wish to consider using inclined coordinates (h_bar, z_bar) to formulate differential equations of motion of the projectile and solving them using MATLAB's ODE solver.

Figure-4: Projectile intercepting drone: notations. (Not to scale).

40957520955000

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TASK-5

(MATLAB: "Integrated knowledge", including projectiles, linear ODEs)

Figure shows a classical pendulum (mass "m", suspended on an massless inextensible rigid link of length "L" at fixed point "O").Suspension point "O" is located above the ground at height "H".To measure the current angular position of the pendulum, the polar angle "theta" is used, measured from the vertical downwards direction, as shown in Figure. Positive direction of the angle "theta" is in the anti-clockwise direction and angle "theta" can have positive or negative value.Pendulum is provided with the initial angular displacement "theta0" (initially positioning mass "m" at point "A") and is then released, without any initial velocity to the mass applied.When the mass "m" reaches the point "B", corresoponding the the angle "theta_B", _B, the pendulum link is cut, enabling mass "m" to freely continue its motion as a projectile, flying to the landing point "C". Determine horizontal coordinate of the point "C" (in metres) in the "xOy" cartesian coordinate system (i.e.distance "X" shown in the Figure): formulate your task using ODEs, solve the problem, using MATLAB, round your result for "X" and enter as a whole number.Ignore air resistance and rotational inertia of the mass "m". Use the following parameters: m=1 kg; L=16 m; H=-22 m; theta0=-170 deg (negative); theta_B=55 deg (positive).

Figure-5: Pendulum, launching its mass as a projectile. (Note: dimensions are not to scale.)

YOUR TASK-5 ANSWER:*Enter your result as an integer number (for example, 123) without trailing spaces.

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TASK-6

(Projectile motion: linear versus non-linear model, coordinate systems, ode45, MATLAB, interp1)

During the experiment, mass m=5 kg was released at point 'A' and travelled distance AB=Z before it hit the inclined flat surface, as shown in Figure. After the perfect impact (with no energy loss) with the incline at point 'B', mass 'm' bounced and continued its flight until it hit the incline again at point 'C'.Assume AIR RESISTANCE with air resistance force is proportional to the product of the squared velocity of the mass and coefficient Cd=0.22 (i.e. F=Cd*v^2) and the following parameters of the experiment: inclination angle alpha=30 deg; Z=4 m. DETERMINE: the distance 'BC' (in m), formulating and solving numerically the associated differential equations of motion for the mass 'm'.

60007542227500Figure-6: Mass "m", bouncing from the inclined surface.

PLEASE, READ ADDITIONAL RECOMMENDATIONS AND ENTER YOUR ANSWER for TASK-6 BELOW:*Round your result up to the integer and select one of the most closely matching (within plus/minus 1) numerical answer.Note: Your correct answer may be discounted if you do not submit a properly working MATLAB script, producing correct answer.

SUBMIT YOUR MATLAB SCRIPT FOR Task-6