Write a matlab function that solves a quadratic equation

Assignment Help Engineering Mathematics

Reference no: EM131372219

Part 1: Solving a quadratic equation

This exercise is similar to what you should have already completed for one of the exercises in Chapter 3 of the MATLAB course notes. Note however, that it is not exactly the same as the format of inputting and outputting values differs from the example in the notes.

Write a MATLAB function that solves a quadratic equation of the form

ax² + bx + c = 0

The syntax of your function should take the form

>> [quadRoots] = Q1 STNO(coeff);

where STNO should be replaced by your student number.

The function should accept one input argument, coeff, a three element vector of the quadratic coefficients in the order a, b, c.

Your function should return one output argument, quadRoots. For the case where two roots are present, this should be a two-element vector of them. For the case where one root is present, the output should be a scalar of that value. For the case where no roots are present, the output should be a scalar set to NaN (not a number, see Section 1.3.3 of the course notes).

It is permissible for coeff and quadRoots to take the form of either row or column vectors.

Code format and error-free execution

Output when two roots are present

Output when one root is present

Output when no roots are present

Part 2: Factorials

The factorial of a positive integer is defined as:

n! = n × (n - 1) × (n - 2) × (n - 3) × . . . × 1

where n! = 1 when n = 0. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120.

The double factorial of a positive odd integer is defined as:

n!! = n × (n - 2) × (n - 4) × . . . × 1

where n!! = 1 when n = 0. For example, 9!! = 9 × 7 × 5 × 3 × 1 = 945.

Write a MATLAB function that computes both the factorial and double factorial for a positive odd integer.

The syntax of your function should take the form

>> [fact,DblFact] = Q2 STNO(n);

where n is the number your wish to find the factorials of, and fact and DblFact are the factorial and double factorial respectively.

Add a few error checks to the input to check that is sensible. (You might need to do a small amount of background research to find a way to verify the input is a odd integer).

Code format and error-free execution

Factorial

Double factorial

Error checking

Part 3: A vertically falling ball

The drag force D(t) as a function of time t acting on an object falling at velocity v(t) is given by

D(t) = c_d1/2ρv(t)² A

where c_d is a dimensionless constant known as the drag coefficient that depends on the geometry of the object (use c_d = 0.47 for a sphere), ρ is the density of air (use ρ = 1.2 kg/m³) and A is the cross-sectional area, which is circular for a ball.

The total force on a falling ball is thus the difference between drag and gravity:

F(t) = c_d1/2ρv(t)²A - mg

where g is the acceleration due to gravity = 9.81 m/s².

The acceleration a(t) is therefore (using Newton's Second Law):

a(t) = (c_dρA/2m)v(t)² - g (1)

Note that all the terms here are constant except for a(t) and v(t).

Because the acceleration is not constant, we cannot use "suvat" equations but can ap- proximate velocities sequentially using:

v_n = v_n-1 + a_nδt (2)

where vn and an approximate the velocity and acceleration at time t_n (= nδt) respectively, and δt is the time step between values (see Annex 1 for derivation). Please use δt = 0.1s for this assignment.

Write a MATLAB function to calculate the time taken t_Thresh (in seconds) for a vertically falling ball of mass m (in kg) and diameter d (in m) to reach a velocity of v_Thresh in the presence of drag assuming the initial velocity is zero. Your function should also return the terminal velocity v_Term and the fraction of the terminal velocity reached at this time (e.g. if v_Term = 50 m/s and v_Thresh = 25 m/s, then the fraction is 0.5).

The syntax for your code should take the form

>> [tThresh,vTerm,vFrac] = Q3.STNO(m,d,vThresh)

where m and d are the mass and diameter of the ball respectively and vThresh is the velocity at the required time.

[Hint: You may download Q3 TEMPLATE.m to use as a starting point. Assume that the mass of the ball falls in the range 5 kg to 20 kg and that the diameter is in the range 5 cm to 50 cm.]

Error free execution

Calculated time

Terminal velocity

Fraction of terminal velocity

Part 4: Bouncing ball

N.B. This part of the assignment is designed to be challenging with very limited help available!

In an ultra-exciting physics experiment, some students plan to launch a bouncy ball from a toy cannon. Assuming that the ball is launched at velocity v (in m/s) and elevation angle θ (in radians measured from the horizontal ground to direction of cannon), write a MATLAB function that plots the trajectory of the ball for the first two bounces on a well-labelled graph. Unlike for Part 3, assume no drag.

Your code should take the form:

>> [d] = Q4 STNO(v,theta,c)

where d is a two-element vector indicating the distances of the first two points of contact with the ground from the cannon, v and theta are the launch velocity and angle as stated above, and c is defined below. An example output is shown in Figure 1 for clarity.

Figure 1: Sample result. In this case, d would be approximately [10.2 17.8].

Assumptions:

- No drag/wind.

- The ball is launched from the ground.

- The ground is perfectly flat and level with no obstacles.

- There is no spin on the ball and no friction between the ball and the surface: hence no change in horizontal velocity after each bounce.

- Vertical velocity changes by a multiplicative factor of -c at the instant of each bounce.

Distance to bounces

Graph (trajectory)

Graph (style, clarity, labelling)

For a final challenge, add two additional inputs to specify the distance from the origin and the height of a thin wall. Your function should then plot the new trajectory including a rebound from the wall. Assume that the bounce between the ball and wall does not affect the magnitude of the velocity (unlike the bounce with the ground).

If you attempt this final part then please submit two functions for Part 4: one without this final calculation as previously ([d] = Q4 STNO(v,theta,c)) and one of the form:

>> [d] = Q4e STNO(v,theta,c,D,H)

where D and H refer to the distance and height of the wall respectively. You should plot the wall on the graph using the MATLAB line function (type help line into the MATLAB command window for more details).

Reference no: EM131372219

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Reviews

len1372219

1/27/2017 12:10:48 AM

Q You have tested your code on a range of input parameters and, where possible, verified your answers with pen and paper or a calculator. This may not always be possible, but at least check that your function outputs are sensible – if your code returns a velocity of one billion m/s then something is probably wrong! Q MATLAB is case-sensitive. Make sure you are consistent with which case you use. For example, if you define velocity in the x-direction as vx initially, and then start using Vx, you will most likely get an error. Q After submission to Blackboard, you have re-downloaded all your files to verify that the correct versions are uploaded and are not corrupted.

1/27/2017 12:10:38 AM

Q Functions work correctly when called from the command window. In Part 1 for example, the code will be marked by typing something similar to “roots = Q1 01234567([1 2 -3])” into the command window (as opposed to pressing F5 or the “run” button from the MATLAB editor). Q Submitted codes are function files requiring inputs and outputs (as opposed to script files – see Section 3.2 of the course notes). Q P-code files will not be marked. All submissions should have a .m extension. Q Each function is entirely self-contained – i.e. does not require any non-built-in func- tions to work correctly.

1/27/2017 12:10:28 AM

If necessary, refer to the Blackboard help page of the UWE website for generic Blackboard submission instructions1 and verify that your upload was successful by downloading an electronic receipt2. It is recommended that you download and re-open your submitted files to make sure the correct versions are uploaded. General assessment guidelines can also be found online3. This includes information on assessment offences. All submitted functions will be tested against undisclosed input parameters. There may be slight discrepancies between any two correct answers due to rounding errors (for example) but full marks will be awarded where answers are within a reasonable tolerance of the exact values and a sound methodology is adopted.

1/27/2017 12:09:56 AM

You are required to complete the following tasks for the Engineering Mathematics MAT- LAB assignment. The assignment is divided into four parts, each worth 25 marks. Please submit code for any part that you have attempted, even if you are unable to complete the entire assignment. Completed MATLAB functions should be submitted via Blackboard before the deadline of 14:00. For each of the four parts, you should prepare a .m file in the given format. Please ensure that your functions exactly follow the format given in the questions (remember, in particular, that MATLAB is case sensitive). Furthermore, each function must be entirely self-contained, i.e. it should not call other functions apart from built-in ones. You can submit your solutions at any time leading up to the deadline. If you resubmit your work, then any earlier versions will be overwritten, but note that all files must be resubmitted if you do this (i.e. not just the files that you have changed since the last attempt). It is recommended that you familiarise yourself with the submission procedure well before the submission date to make sure you understand the system.

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