Answer:
The uncertainty in the average speed is 0.134 meters per second.
Step-by-step explanation:
Let be
the average speed function, we calculate the uncertainty in the average speed by total differentials, which is in this case:
![\Delta v = \frac{\partial v}{\partial x}\cdot \Delta x+\frac{\partial v}{\partial t}\cdot \Delta t](https://tex.z-dn.net/?f=%5CDelta%20v%20%3D%20%5Cfrac%7B%5Cpartial%20v%7D%7B%5Cpartial%20x%7D%5Ccdot%20%5CDelta%20x%2B%5Cfrac%7B%5Cpartial%20v%7D%7B%5Cpartial%20t%7D%5Ccdot%20%5CDelta%20t)
Where:
- Uncertainty in the average speed, measured in meters per second.
- Partial derivative of the average speed function with respect to distance, measured in
.
- Partial derivative of the average speed function with respect to time, measured in meters per square second.
- Uncertainty in distance, measured in meters.
- Uncertainty in time, measured in seconds.
Partial derivatives are, respectively:
, ![\frac{\partial v}{\partial t} = - \frac{x}{t^{2}}](https://tex.z-dn.net/?f=%5Cfrac%7B%5Cpartial%20v%7D%7B%5Cpartial%20t%7D%20%3D%20-%20%5Cfrac%7Bx%7D%7Bt%5E%7B2%7D%7D)
Then, the total differential expression is expanded as:
![\Delta v = \frac{\Delta x}{t}-\frac{x\cdot \Delta t}{t^{2}}](https://tex.z-dn.net/?f=%5CDelta%20v%20%3D%20%5Cfrac%7B%5CDelta%20x%7D%7Bt%7D-%5Cfrac%7Bx%5Ccdot%20%5CDelta%20t%7D%7Bt%5E%7B2%7D%7D)
If we get that
,
,
and
, the uncertainty in the average speed is:
![\Delta v = \frac{2.3\,m}{6.3\,m}-\frac{(6.1\,m)\cdot (1.5\,s)}{(6.3\,s)^{2}}](https://tex.z-dn.net/?f=%5CDelta%20v%20%3D%20%5Cfrac%7B2.3%5C%2Cm%7D%7B6.3%5C%2Cm%7D-%5Cfrac%7B%286.1%5C%2Cm%29%5Ccdot%20%281.5%5C%2Cs%29%7D%7B%286.3%5C%2Cs%29%5E%7B2%7D%7D)
![\Delta v = 0.134\,\frac{m}{s}](https://tex.z-dn.net/?f=%5CDelta%20v%20%3D%200.134%5C%2C%5Cfrac%7Bm%7D%7Bs%7D)
The uncertainty in the average speed is 0.134 meters per second.