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MATH 3040 Topics in Mathematics 1: Theoretical Mechanics
Assignment
Particle Dynamics in 2D and 3D
This is an individual assessment piece. It is expected that students will submit results of their own working and
that no solution has been devised by a third party. Penalties apply.
This assignment is worth 30% of the final mark for this course.
Date Due: May 13, 2022, 5pm
Make sure to relate on a coversheet any information about help given/received for any of the questions you
attempted, who received/gave help, and to reference any sources other than the textbook for the course.
Assignment Question (50 marks)
A projectile is fired from the origin O with velocity v0 and at an angle α to the horizontal through a resistive
atmosphere, to land on the same horizontal plane some distance away. During its flight the projectile experiences an
air resistance that is proportional to the projectile’s velocity, but which diminishes exponentially with height. That is,
the projectile experiences a resistance force which follows the empirical law
F(v, z) = −
[
be−z/h
]
v,
where b is a (constant) resistance coefficient, h is a (constant) length scale, and v is the projectile’s velocity at time
t. The path taken by the projectile is given by the solution of the dynamic equation based on Newton’s second law of
motion:
mr¨(t) = −mgj+ F(r˙, z). (1)
It has been found that a projectile passing through a so-called “thinning” atmosphere (such as one represented by Eq
(1)) achieves a longer range than is predicted by the exact result of a projectile’s path through a uniform atmosphere
with an air resistance coefficient of b, under otherwise identical conditions.
This assignment concerns a full description of the projectile motion and trajectory including the latter’s depen-
dence on external (b, h) and initial conditions (α, |v0|). Your mathematical analysis of the motion should include, but
possibly not be limited to, the following steps.
(A) Establish an appropriate coordinate system to represent the dynamics.
(B) Describe the respective components of the motion, reflecting on any interdependencies and their consequences.
(C) Produce numerical solutions of the parametric equations of motion for the projectile’s trajectory, assuming suit-
able numerical values of the initial conditions.
(D) If possible, derive a closed form expression for the solution of the vertical component of the equation of motion,
and compare this expression with your numerical solution determined in Part (C).
(E) Determine the angle of projection α that would result in the maximum range of the projectile and determine the
latter, assuming the same initial speed. Compare these with the corresponding angle and range obtained for a
projectile passing through a uniform atmosphere.
Note: no loss of marks will result if you are unable to determine an exact solution in Part (D). However, a solution
will generate bonus marks that will be counted toward the final course mark.
1
Expectations of your submission.
[i] Address each part of the problem (as suggested above) in order and insert figures where appropriate. Answers
should be in sentence form. Do not use question or item numbers to label your responses. Rather, your report
on each aspect and others should flow — it should tell a “story”.
[ii] It is expected (but not obligatory) that you will use Matlab for your numerical work. However, whatever
software resources you do choose to use, the coding should be of your own development.
[iii] In your submission you should provide a print out of your working numerical code. Alternatively, you may
submit your .m file. Marks (30%) will be deducted if no numerical code is provided. Please be prepared, if
requested, to demonstrate that the code does indeed work and does produce the results shown in your report.
[iv] Graphs of the projectile’s trajectory, or other graphical information, should be clearly presented and well anno-
tated: axes should be clearly labeled; In the case of multiple curves, a legend should be provided; all figures
must be numbered and accompanied by a figure caption describing the figure content and explaining the condi-
tions under which the graphs were generated.
[v] Any numerical scheme, such as an integration scheme, or an interpolation scheme, or a root finding scheme
(to estimate the value of the optimum angle α), should be clearly, but very briefly, introduced and explained
(including any software packages, such as Matlab’s ode45 solver).
[vi] A written description of the problem and its solution, including any consequential findings, should be included
under the heading “Discussion”. Marks (up to 30%) will be deducted if no Discussion section is found or if the
discussion is inadequate.
[vii] Your assignment may be typeset, but a very clear and neat handwritten report is also acceptable. For typed
assignments, you can insert handwritten mathematical expressions (e.g., as pictures in the text). Typed math-
ematical expressions need to be notationally correct. If using Word to typeset your assignment, you may use
Equation Editor for inline mathematical expressions and for equations. Marks will be deducted for poorly-
presented mathematical expressions. Marks will not be deducted for neatly handwritten expressions.
[viii] As stated above, this is an individual assessment. You may discuss the problem with your colleagues, but
the work must demonstrably be your own. The numerical code, the results and the report, particularly the
Discussion section, must be your own work.
