Source: Final Exam Practice Problems and Solutions
Tags: uncertainty, error propagation, partial derivatives, kinetic energy uncertainty, Bragg's law, forward finite difference, numerical differentiation, momentum calculation, ENGR 216, PHYS 216
Uncertainty propagation lets you calculate how measurement errors compound when plugged into a formula. You take partial derivatives of the formula with respect to each measured variable, multiply by that variable's uncertainty, and combine in quadrature. Forward finite differences give you a numerical approximation of a derivative (velocity, acceleration) from discrete position-time data.
Uncertainty (error)
The range within which the true value of a measurement is expected to fall, expressed as ± some value.
Propagation of uncertainty
The method for determining how uncertainties in measured quantities carry through to a calculated result. Uses partial derivatives and root-sum-of-squares (RSS).
Partial derivative
The derivative of a multivariable function with respect to one variable, treating all others as constants. Central to the propagation formula.
Root-sum-of-squares (RSS)
The method of combining individual uncertainty contributions: square each, sum them, take the square root. Also called "adding in quadrature."
Forward finite difference
A numerical method for approximating a derivative at a point using the value at that point and the next point: f'(x) ≈ [f(x + h) − f(x)] / h.
Nominal value
The calculated result using the best (central) measured values, before uncertainty is applied.
For a quantity Q that depends on measured variables x₁, x₂, ..., xₙ:
δQ = √[ (∂Q/∂x₁ · δx₁)² + (∂Q/∂x₂ · δx₂)² + ... + (∂Q/∂xₙ · δxₙ)² ]
The procedure is always the same:
Write out the formula for Q
Take the partial derivative of Q with respect to each measured variable
Multiply each partial derivative by the uncertainty in that variable
Square each term, sum them, take the square root
Given: m = 10.0 ± 0.5 kg, v = 3.0 ± 0.2 m/s, KE = ½mv²
∂KE/∂m = ½v² = ½(3.0)² = 4.5
∂KE/∂v = mv = (10.0)(3.0) = 30
δKE = √[(4.5 × 0.5)² + (30 × 0.2)²] = √[5.0625 + 36] = √41.0625 = 6.41 J
Given: V₁ = 125 ± 5 mL, V₂ = 25 ± 2 mL, V_remaining = V₁ − V₂
∂V/∂V₁ = 1, ∂V/∂V₂ = −1
δV = √[(1 × 5)² + (−1 × 2)²] = √[25 + 4] = √29 = 5.39 mL
Note that when you subtract two measurements, the uncertainties still add in quadrature. They never cancel.
Given: r = 3.2 ± 0.15 cm, V = (4/3)πr³
∂V/∂r = 4πr² = 4π(3.2)² = 128.68
δV = 128.68 × 0.15 = 19.3 cm³
Single-variable case: the partial derivative is just the ordinary derivative.
Given: d = 0.025 ± 0.0020 nm, θ = 25 ± 1.0°, nλ = 2d sin(θ), n = 1
Nominal: λ = 2(0.025) sin(25°) = 0.0211 nm
∂λ/∂d = 2 sin(θ) = 2 sin(25°) = 0.8452
∂λ/∂θ = 2d cos(θ) = 2(0.025) cos(25°) = 0.04532
Important: convert the angle uncertainty to radians before plugging in. 1.0° = 0.01745 rad.
δλ = √[(0.8452 × 0.0020)² + (0.04532 × 0.01745)²] = √[2.857 × 10⁻⁶ + 6.257 × 10⁻⁷] = 0.0019 nm
Final answer: λ = 0.021 ± 0.0019 nm
To approximate velocity from position-time data:
v(tᵢ) ≈ [x(tᵢ₊₁) − x(tᵢ)] / [tᵢ₊₁ − tᵢ]
Then momentum p = m × v.
Given: m = 2.0 kg, positions in cm at 0.1 s intervals.
Time (s) | Position (cm) |
|---|---|
0.1 | 1.0 |
0.2 | 3.9 |
0.3 | 9.1 |
0.4 | 15.8 |
0.5 | 24.9 |
At t = 0.1 s: v = (3.9 − 1.0) cm / 0.1 s = 29 cm/s = 0.29 m/s, p = 2.0 × 0.29 = 0.58 (kg·m)/s
At t = 0.2 s: v = (9.1 − 3.9) / 0.1 = 52 cm/s = 0.52 m/s, p = 2.0 × 0.52 = 1.04 (kg·m)/s
At t = 0.3 s: v = (15.8 − 9.1) / 0.1 = 67 cm/s = 0.67 m/s, p = 2.0 × 0.67 = 1.34 (kg·m)/s
At t = 0.4 s: v = (24.9 − 15.8) / 0.1 = 91 cm/s = 0.91 m/s, p = 2.0 × 0.91 = 1.82 (kg·m)/s
General uncertainty propagation: δQ = √[ Σ (∂Q/∂xᵢ · δxᵢ)² ]
Kinetic energy: KE = ½mv²
Volume of a sphere: V = (4/3)πr³
Bragg's Law: nλ = 2d sin(θ)
Forward finite difference: f'(xᵢ) ≈ [f(xᵢ₊₁) − f(xᵢ)] / h
⚠️ Always convert angle uncertainties to radians before using them in propagation with trig functions.
⚠️ Subtraction does not reduce uncertainty. The uncertainties still add in quadrature.
⚠️ For forward finite difference, you cannot compute the derivative at the last data point (there is no "next" value).
⚠️ Watch your unit conversions, especially cm to m when computing momentum.
Q: If Q = ½mv² and m = 10.0 ± 0.5 kg, v = 3.0 ± 0.2 m/s, what is δKE?
A: 6.41 J. Take partial derivatives ∂KE/∂m = ½v² and ∂KE/∂v = mv, multiply each by its uncertainty, combine in quadrature.
Q: Why does subtracting two measurements not reduce the uncertainty?
A: Because uncertainties are always squared before summing (RSS). The sign of the partial derivative becomes positive when squared, so addition and subtraction produce the same uncertainty formula.
Q: What is the key limitation of the forward finite difference method?
A: You lose the derivative estimate at the final data point, and the approximation is only first-order accurate (error proportional to the step size h).
Q: When propagating uncertainty through Bragg's Law, what common mistake can double your uncertainty in λ?
A: Forgetting to convert degrees to radians for δθ. The partial derivative ∂λ/∂θ = 2d cos(θ) gives a result in radians, so the uncertainty must also be in radians.
error propagation, uncertainty analysis, RSS, root sum of squares, quadrature, partial derivatives, measurement error, kinetic energy error, Bragg diffraction, X-ray crystallography, forward difference, backward difference, central difference, numerical differentiation, finite difference method, momentum from position data, ENGR 216 Module 2, ENGR 216 Module 3