This lab will probably take us two lab periods. The assignment is therefore due at the beginning of lecture on Friday, 23 Feb. Your solution should include all codes and plots, as well as written responses where appropriate. You may use your own ode solver or the generic ODE solver & codes I have posted off of the course outline.

In this lab we'll investigate projectile motion much like you studied in PHYS 211; however, we'll also consider the effects of air resistance. For these problems the Euler method is sufficient to obtain precise solutions; however, you may use a higher order method if you so choose. You may also chose to investigate the differences between methods in arriving at your solutions (e.g., compare Runge-Kutta 4 to Euler).

Throughout this lab we'll consider a few simplifying assumptions. All of our projectiles are fired from a planar surface, in the presence of a uniform gravitational field. We'll also assume that the atmosphere is motionless, isobaric and isothermal. If you're having no trouble with MATLAB programming, you are encouraged to relax some of those assumptions (see 8-10). I expect that everyone should, using my codes if necessary, be able to work through 1-7 by the due date.

From PHYS 211 you should recall that, neglecting air resistance, projectiles launched from/to a plane have identical ranges (r) for complimentary angles of launch. That is, r_θ = r_90°-θ, where r(θ) = v_o² sin 2θ / g. This result isn't too practical for actual weapons systems — mortars typically cannot be depressed to less than 65° and howitzers (modern cannons) cannot be elevated to more than 35° — so our "gun" will have to be somewhat of a hypothetical weapon. We'll use the projectile equations of motion we developed in class for this lab. Keep in mind the points we discussed in class.

Our gun will be a customized 105 mm howitzer firing a high explosive (HE) round. We'll assume the following characteristics for this gun:

• muzzle velocity: 310 m/s (fixed and just subsonic)
• blast radius: 30 m
• elevation angle range: 5° – 60° (this is the only unrealistic characteristic, it's actually -5° – 35°)
• projectile mass: 15 kg
• drag coefficient: 6×10^-4 kg/m

Given these characteristics, complete the following tasks:

1. select two angles and their complements, within the elevation limits of our gun;
2. verify graphically that the ranges for the pairs are identical in vacuum;
3. verify numerically that these results are within half a blast radius of the theoretical range in vacuum;
4. allow for air resistance and graphically compare the results with the same angles as above;
5. allow for air resistance and find the range for θ = 35°;
6. find another angle φ > θ such that r_θ = r_φ when air resistance is a factor;
7. verify numerically that these results are within 1/10-th of a blast radius of one another.

Those students who are feeling comfortable with MATLAB and desire a greater challenge are encouraged to do the following problems. We're going to consider two changes to our modeling of the physics in the problem. The first is the fact that our assumption F_g = -mg was only an approximation to the force due to gravity. We know that this force changes with altitude, so that F_g = -GM_em/r² is a better model. We further assumed that F_drag = -bv², which is a good approximation of the drag at the firing position of the gun. However, since the shell reaches considerable altitude — the atmosphere is not isobaric, nor isothermal — a better approximation would be F^*_drag = ρ/ρ_o F_drag (y = 0), where ρ(y) = ρ_o ( 1 - ay/T_o)^α. All the other assumptions remain intact. For the following problems you should provide code(s), graphical and or numerical evidence that you've verified your results.

8. A practice target is situated 6 km from the firing point. Find the firing angle(s) that cause a shell to "hit" the target (impact within 1/10-th of a blast radius) using the new model. Assume α = 7/2, a = 6.5×10^-3 K/m, and T_o = 300 K.
9. Considering the problem above, compare the effects of each of the new model components with the model used in §§4-7. We can do this by variously setting a = 0 (isobaric/isothermal atmosphere) and r = r_e (uniform gravitational field). Which assumption, a = 0 or r = r_e, is physically more sound? Justify your response.
10. Automate your approach(es) to finding solutions to any of the problems 1-9.