Reference no: EM132528251

Computer Lab Assignment

Question 1. Equation of Motion
Develop the equations of motion of this projectile motion in Figure 1. Show all steps including a free-body-diagram (include all dragforce components).

Question 2. Analytical Solutio
Linearization: Obtain the linearized version of the non-linear equations of motion defined in part 1 by neglecting drag.
Using thislinearized equation of motion from(i), derive for the analytical solution for the position, velocity and acceleration as functions of time.
Determine the time needed for the projectile to reach the groundy(t)=0and the corresponding distance travelled (x-coordinate).

Question 3. Finite Differences
i. Derive the finite difference equation for the projectile, starting from the non-linear equation of motion obtained in part 1. Show all steps.
ii. In addition, define and justify how finite differences is initialised.
iii. Show your MATLAB code for the implementation of the finite difference problem from (i) and (ii).

Question 4. Discretization and Convergence
The resolution of your solution is completely dependent on time-step dt and can increase indefinitely as dt→0. Because of this, in the practical application of computational resources, we are only interested indt which will provide results that do not change/benefit greatly from any further increase in resolution.

Demonstrate and justify your choice for time-step dt and the total simulation time. To start, set dt=1[s] and choose a total time that allows for the project to reach the ground. Plot the position (x,y) obtained from the Analytical and Numerical Solutionneglecting drag force on the same figure.(Note:one possible way toneglect drag forceis by simply setting C_d=0)
Comment on the resolution of results, is this dt=1 [s] adequate?
If not, change dt until sufficient resolution is achieved.To assess this, compare the landing position x_(y=0) for at least five different dt against the analytical x_(y=0) from Part2, using the table provided below. Show the plot of the position (x,y) with your chosen value for dt.

Question 5. Analytical vs. Numerical (neglecting drag force)
Using the linear analytical solution from part2 and the numerical solution from part 3neglecting drag force (e.g., can be done by simply setting C_d=0)
Show a separate plot for each of the following variables with respect to time:vertical position, the magnitude of velocity, and magnitude of acceleration.In each plot, for each variable, display boththe Analytical and Numerical results for comparison.
[suggestion: use subplots to generate three separate plots for each variable all in one figure. Plot from time t=0 to the time taken to reach the ground, i.e. when t_(y=0).]
Comment and provide an explanation for the agreement (or, disagreement) of analytical vs. numerical solution (neglecting drag force) results.

Question 6. Numerical Investigation
In this section,the effect of air resistance (drag force) will be investigated numerically by using only the numerical solution (i.e., finite difference)from part 3to solve the case where drag is neglected (as done similarly in part 5) and the case where drag is included.

Plot the position (x,y) of the projectile for the results withandwithout drag force in the same figure for comparison.
Also, show a separate plot for each of the following variables with respect to time:vertical position, the magnitude of velocity, and magnitude of acceleration. In each plot, for each variable, display both the results withandwithoutforce in the same figure for comparison.

Comment on the characteristics observed and provide an explanation for the agreement (or, disagreement) of the numerical results with and without drag force.
Suggestion, consider the following:
How well does the linear assumption compare against the fully non-linear equation?
Is drag relevant in the motion?

Attachment:- Computer Lab Assignment.rar