A parachutist with a mass of $68.1 \text{ kg}$ jumps out of a stationary hot air balloon. The drag coefficient is $c = 12.5 \text{ kg/s}$. Compute velocity prior to opening the chute at $t = 2 \text{ s}$. Acceleration due to gravity is $g = 9.81 \text{ m/s}^2$.

The analytical solution for velocity is:

The rate of decay of a radioactive substance is proportional to the amount of substance present: $\frac{dN}{dt} = -kN$. Given an initial amount $N_0 = 100 \text{ g}$ and a decay constant $k = 0.05 \text{ day}^{-1}$, compute the amount remaining after $10 \text{ days}$ using the exact analytical model.

An unforced, undamped spring-mass system is modeled by $m\frac{d^2x}{dt^2} + kx = 0$. Given $m = 2 \text{ kg}$, $k = 50 \text{ N/m}$, initial displacement $x(0) = 0.5 \text{ m}$, and initial velocity $v(0) = 0 \text{ m/s}$, determine the displacement at $t = 1 \text{ s}$.

An engineering team measures the mid-span deflection of a bridge under a test load. The true theoretical deflection is $15.0 \text{ mm}$.

Team A's measurements: $14.8 \text{ mm}, 14.9 \text{ mm}, 15.1 \text{ mm}, 15.0 \text{ mm}$

Team B's measurements: $16.5 \text{ mm}, 16.5 \text{ mm}, 16.6 \text{ mm}, 16.5 \text{ mm}$

Analyze the accuracy and precision of both teams.

Three batches of concrete cylinders are tested for their 28-day compressive strength. The design target (true value) is $30 \text{ MPa}$.

Batch X: $29.8, 30.2, 30.1, 29.9 \text{ MPa}$ Batch Y: $25.1, 35.0, 28.5, 31.4 \text{ MPa}$ Batch Z: $34.8, 35.1, 35.0, 34.9 \text{ MPa}$

Evaluate the accuracy and precision.

The true value of $\pi$ is approximately $3.14159$. A student estimates $\pi$ as $22/7 \approx 3.142857$. Calculate the true error and true percent relative error.

The derivative of a function $f(x) = -0.1x^4 - 0.15x^3 - 0.5x^2 - 0.25x + 1.2$ at $x = 0.5$ is true value $-0.9125$. A numerical method estimates the derivative to be $-0.9025$. Calculate the true error and the true percent relative error.

In an iterative numerical method to find a root, the estimate in iteration 2 is $0.5$, and the estimate in iteration 3 is $0.5671$. Calculate the approximate percent relative error.

Approximate $f(x) = e^x$ at $x_{i+1} = 1$ using a zero-order Taylor series based at $x_i = 0$. True value is $e^1 \approx 2.71828$.

Approximate $f(x) = e^x$ at $x = 1$ using a second-order Taylor series based at $x = 0$. True value is $2.71828$.

Approximate the function $f(x) = \cos(x)$ at $x_{i+1} = \frac{\pi}{3}$ using a zero-order and first-order Taylor series expansion based on the base point $x_i = \frac{\pi}{4}$. Compute the true percent relative error for each approximation.

Given: True value $f(\frac{\pi}{3}) = \cos(\frac{\pi}{3}) = 0.5$ $h = x_{i+1} - x_i = \frac{\pi}{3} - \frac{\pi}{4} = \frac{\pi}{12}$

Given two measurements with their associated estimated errors: $x_1 = 100 \pm 2$ $x_2 = 50 \pm 1$

Determine the maximum possible error in the sum $y = x_1 + x_2$.

Given the sides of a rectangle $w = 10 \pm 0.1 \text{ m}$ and $l = 20 \pm 0.2 \text{ m}$, find the maximum error in the calculated area $A = w \times l$.

The deflection of a beam is given by $y = \frac{FL^3}{3EI}$. If the measured force is $F = 100 \text{ N}$ with an error of $\Delta F = \pm 5 \text{ N}$, and the length is $L = 2 \text{ m}$ with an error of $\Delta L = \pm 0.02 \text{ m}$. Assume $E$ and $I$ are exact, where $3EI = 1000 \text{ N}\cdot\text{m}^2$. Use the general first-order error propagation formula to estimate the maximum error in deflection.