Calculate the Uncertainty of Velocity

Error Propagation Formula:

\[ \Delta v = \sqrt{\left(\frac{\Delta d}{d}\right)^2 + \left(\frac{\Delta t}{t}\right)^2 - \frac{2 \text{cov}(d,t)}{d \cdot t}} \times v \]

where \( v = \frac{d}{t} \)

Distance (d):

Uncertainty in Distance (Δd):

Time (t):

Uncertainty in Time (Δt):

Covariance (cov(d,t)):

m·s

Velocity (v):

Uncertainty in Velocity (Δv):

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1. What is Error Propagation?

Error propagation refers to how measurement uncertainties in input quantities affect the uncertainty of a calculated result. When calculating velocity (v = d/t), the uncertainties in distance (Δd) and time (Δt) propagate through the calculation to give an uncertainty in velocity (Δv).

2. How Does the Calculator Work?

The calculator uses the error propagation formula:

\[ \Delta v = \sqrt{\left(\frac{\Delta d}{d}\right)^2 + \left(\frac{\Delta t}{t}\right)^2 - \frac{2 \text{cov}(d,t)}{d \cdot t}} \times v \]

Where:

\( \Delta v \) — Uncertainty in velocity (m/s)
\( \Delta d \) — Uncertainty in distance (m)
\( d \) — Distance (m)
\( \Delta t \) — Uncertainty in time (s)
\( t \) — Time (s)
\( \text{cov}(d,t) \) — Covariance between distance and time measurements (m·s)
\( v \) — Calculated velocity (d/t) (m/s)

Explanation: The formula accounts for how uncertainties in the input measurements combine to affect the final result, including any correlation between the measurements (through the covariance term).

3. Importance of Uncertainty Calculation

Details: Calculating uncertainty is crucial for understanding the reliability of experimental results and for comparing measurements with theoretical predictions or other experiments.

4. Using the Calculator

Tips: Enter all values in SI units (meters for distance, seconds for time). The covariance term can be set to zero if distance and time measurements are independent. All input values must be positive (except covariance which can be negative).

5. Frequently Asked Questions (FAQ)

Q1: What if I don't know the covariance?
A: If distance and time measurements are independent, covariance is zero. This is often the case unless measurements are somehow correlated.

Q2: Why does the formula have a subtraction term?
A: The subtraction accounts for correlation between measurements. If measurements are positively correlated, this reduces the overall uncertainty.

Q3: Can this be used for other calculations?
A: This specific formula is for v = d/t. Different formulas apply to other mathematical operations (addition, multiplication, etc.).

Q4: What if my uncertainties are very small?
A: The calculator works for any uncertainty values, but very small uncertainties relative to the measurements will result in very small percentage uncertainties in the result.

Q5: How accurate is this uncertainty calculation?
A: This is a first-order approximation that works well when uncertainties are small compared to the measurements. For large uncertainties, more complex methods may be needed.