Question

Find the uncertainty in a calculated average speed from the measurements of distance and time. Average speed depends on distance and time according to this function v(t,x) = x/t. Your measured distance and time have the following values and uncertainties x = 6.1 meters, 2.3 meters and t = 6.3 seconds and 1.5 seconds. What is the uncertainty in the average speed, ? Units are not needed in your answer.

Answer 1

The uncertainty in the average speed is 0.134 meters per second.

Explanation:

Let be
`v(t, x) =(x)/(t)` the average speed function, we calculate the uncertainty in the average speed by total differentials, which is in this case:

`\Delta v = (\partial v)/(\partial x)\cdot \Delta x+(\partial v)/(\partial t)\cdot \Delta t`

Where:

`\Delta v` - Uncertainty in the average speed, measured in meters per second.

`(\partial v)/(\partial x)` - Partial derivative of the average speed function with respect to distance, measured in
`s^(-1)`.

`(\partial v)/(\partial t)` - Partial derivative of the average speed function with respect to time, measured in meters per square second.

`\Delta x` - Uncertainty in distance, measured in meters.

`\Delta t` - Uncertainty in time, measured in seconds.

Partial derivatives are, respectively:

`(\partial v)/(\partial x) = (1)/(t)`,
`(\partial v)/(\partial t) = - (x)/(t^(2))`

Then, the total differential expression is expanded as:

`\Delta v = (\Delta x)/(t)-(x\cdot \Delta t)/(t^(2))`

If we get that
`\Delta x = 2.3\,m`,
`t = 6.3\,s`,
`x = 6.1\,m` and
`\Delta t = 1.5\,s`, the uncertainty in the average speed is:

`\Delta v = (2.3\,m)/(6.3\,m)-((6.1\,m)\cdot (1.5\,s))/((6.3\,s)^(2))`

`\Delta v = 0.134\,(m)/(s)`

The uncertainty in the average speed is 0.134 meters per second.