$1.$ Using the Euler method, create a code to model the trajectory of a projectile with a known initial position, speed, and angle of inclination to the

horizontal, $\backslash$theta $.$ Extra marks will be awarded for:

$($a$)$ The quality, conciseness and readability of your code.

$($b$)$ The flexibility of your code $($i$.$e$.$ how easy is it to change variables?$).$

$($c$)$ The quality of your comments.

$($d$)$ How well it works!

$2.$ Using your code, and ignoring drag forces, determine how far a projectile

launched from ground level would travel for an initial velocity of $10$ m$/$s

at an angle of i$)60$ degrees to the horizontal and ii$)30$ degrees to the

horizontal. Extra marks will be awarded for:

$($a$)$ A high$-$quality, well$-$labelled plot of the projectiles$$ motion.

$($b$)$ A reasoned discussion of how well the value obtained agrees with

your expectations.

$($c$)$ A discussion of how varying the timestep dt affects your results.

$($d$)$ Bonus work I: Repeat the above using Velocity Verlet or another

suitable method. How do your answers differ?

$3.$ Extend your model to simulate a ball bouncing on a flat, solid surface.

Use the simple $$hard$-$sphere$$ model taught in the lectures. For the case of

a steel ball in air, model a particle being dropped vertically on the surface

and:

$($a$)$ Adjust the timestep used and comment on how it affects your results.

In particular, see what happens at too large a timestep.

$1$

$($b$)$ Use your observations to choose a sensible timestep to use. Justify

your choice.

$($c$)$ Bonus work II: Add a drag model to your code. Calculate the

terminal velocities of a $5$ mm diameter nylon sphere and a $5$ mm

diameter steel sphere in air, water and glycerine. Comment on the

results obtained. Compare your results to theoretical values for each

of the cases tested. Comment on your results.

Note: It is up to you to choose sensible values for all relevant particle

and fluid properties, and to justify these choices in your answer.