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Masja [62]
3 years ago
9

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.
Mathematics
1 answer:
maks197457 [2]3 years ago
8 0

Answer:

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

Step-by-step explanation:

Let be v(t, x) =\frac{x}{t} 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

Where:

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

\frac{\partial v}{\partial x} - Partial derivative of the average speed function with respect to distance, measured in s^{-1}.

\frac{\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:

\frac{\partial v}{\partial x} = \frac{1}{t}, \frac{\partial v}{\partial t} = - \frac{x}{t^{2}}

Then, the total differential expression is expanded as:

\Delta v = \frac{\Delta x}{t}-\frac{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 = \frac{2.3\,m}{6.3\,m}-\frac{(6.1\,m)\cdot (1.5\,s)}{(6.3\,s)^{2}}

\Delta v = 0.134\,\frac{m}{s}

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

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