calculate the speed with uncertainty: 1.01+-.0005 and .9+-.01

To calculate the speed with uncertainty, we need to divide the two quantities and propagate their uncertainties. The formula for propagating uncertainties when dividing two quantities is:

\[
\frac{\Delta v}{v} = \sqrt{\left(\frac{\Delta x}{x}\right)^2 + \left(\frac{\Delta t}{t}\right)^2}
\]

Where:
- \( v \) is the speed, \( v = \frac{x}{t} \)
- \( \Delta v \) is the uncertainty in speed
- \( x = 1.01 \pm 0.0005 \) (distance with uncertainty)
- \( t = 0.9 \pm 0.01 \) (time with uncertainty)

Now, let's calculate the speed and its uncertainty.

1. The speed itself is:

\[
v = \frac{1.01}{0.9} \approx 1.1222 \, \text{units}
\]

2. For the uncertainty:

\[
\frac{\Delta v}{v} = \sqrt{\left(\frac{0.0005}{1.01}\right)^2 + \left(\frac{0.01}{0.9}\right)^2}
\]

Now I'll compute the values.The calculated speed is approximately \( v = 1.122 \, \text{units} \), with an uncertainty of \( \Delta v \approx 0.0125 \, \text{units} \).

Thus, the speed with uncertainty is:

\[
v = 1.122 \pm 0.0125 \, \text{units}
\]

Would you like further details or have any questions? Here are some related questions:

1. How is uncertainty propagation handled in multiplication and division?
2. What are the typical sources of uncertainty in physical measurements?
3. Can uncertainty be reduced by taking more measurements?
4. How would the uncertainty change if the time uncertainty was reduced to 0.005?
5. What is the significance of significant figures in reporting uncertainties?

**Tip:** Always express uncertainty with the same number of decimal places as the measurement it refers to!

Learn how to calculate speed with uncertainty using values 1.01±0.0005 (distance) and 0.9±0.01 (time). This guide explains uncertainty propagation, including the formula for error propagation in division. Suitable for undergraduate physics or math students.

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