Homework 7: Measurement Error Propagation in Projectile Motion

We aim to compute the expected range of a projectile and the uncertainty in that range due to multiple sources of error, including air resistance, atmospheric variability, and launch conditions.

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🎯 Problem Setup

Consider a WWI-era shell with the following properties:

• Mass: 50 kg
• Diameter: 155 mm
• Length: 1.0 m

Assume that it flies through air without tumbling, i.e., it maintains a stable orientation (as argued in class, this may not always be the case).

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💨 Air Resistance Model

In the absence of air resistance, the projectile range is:

$$D = \frac{v_0^2}{g} \sin(2\theta)$$

However, due to air drag, this range is significantly reduced. The drag force is given by:

$$\vec{F}_D = -\frac{1}{2} c \rho A v \vec{v}$$

where:

• $c$: drag coefficient
• $\rho$: air density
• $A$: cross-sectional area
• $v$: speed

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📊 Uncertainty and Distributions

The following quantities are uncertain and have known distributions:

Quantity	Symbol	Distribution	Parameters
Air density	$\rho$	Uniform	[1.0, 1.293] kg/m³
Drag coefficient	$c$	Normal	mean = 0.45, std = 0.005
Initial velocity	$v_0$	Triangular	[999, 1000, 1001] m/s
Launch angle	$\theta$	Normal	mean = your choice (e.g., 45°), std = 2°

Note: Remember to convert degrees to radians before calculations.

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📌 Task

1. Simulate the projectile motion using numerical solver from scipy library (e.g., solve_ivp).
2. Use error propagation, which we learned in class Error propagation, to compute error in projectile range.

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📈 Output

• Report the expected projectile range $D$ and its standard deviation $\sigma_D$.
• Find the optimum angle for the projectile range. Withouth air resistance it is 45°, but with air resistance, it is not.

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🔧 Assumptions

• Constant gravitational acceleration: $g = 9.81{m/s}^2$.
• Shell is axisymmetric and drag is applied on cross-sectional area: $A = \pi \left( \frac{d}{2} \right)^2$

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💬 Questions to Reflect On

• How sensitive is the range to each parameter?
• Which source of uncertainty contributes the most?

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Deliverable: Upload your code and a brief report with simulation results, figures, and your discussion.