Calculating Speed and Uncertainties in Measurements
How do we calculate the average speed of a toy car?
By using a 2-meter stick marked in millimeters and a stopwatch, you measure the distance and time taken by the toy car to travel a certain interval. Given the data of 162.0cm distance and time data of 2.95 s, how do we determine the average speed of the toy car?
What are the absolute and relative uncertainties of the distance and time measurements?
What are the errors in the measurements of distance and time, and which one is more uncertain? How do we calculate the relative and absolute uncertainties?
Why is it necessary to calculate relative uncertainties in a speed estimation?
What is the importance of considering relative uncertainties in measurements? How does it help in determining the overall error in a derived magnitude?
Answer:
Explanation: The average speed of a body is defined as the ratio between total distance and total time:
$$v = \\frac{dx}{dt}$$
$$v = \\frac{162.0}{2.95}$$
$$v = 54.9 m/s$$
The absolute errors (uncertainties) of the distance and time measurements as measured with instruments are the errors of the instruments:
$$\Delta x = 0.1 cm$$
$$\Delta t = 0.01 s$$
Relative errors (uncertainties) are the absolute errors between the measured value:
$$Er = \\frac{Δx}{x}$$
$$Er = \\frac{0.1}{162.0}$$
$$Er = 6.2 x 10^{-4} \text{ length}$$
$$Er = \\frac{0.01}{2.95}$$
$$Er = 3.4 x 10^{-3} \text{ time}$$
The most uncertain measure is the time to have a greater relative error.
Let's calculate the relative speed error:
$$\\frac{Δv}{v} = \\frac{dv}{dx} Δx + \\frac{dv}{dt} Δt$$
$$\\frac{dv}{dx} = \\frac{1}{t}$$
$$\\frac{dv}{dt} = x (-\\frac{1}{t^2})$$
$$Er = \\frac{Δv}{v} = \\frac{1}{t} Δx + \\frac{x}{t^2} Δt$$
$$Er = \\frac{0.1}{2.95} + \\frac{162.0}{2.95^2} x 0.01$$
$$Er = 0.034 + 0.19$$
$$Er = 0.22$$
We can observe that the relative error of time is much higher than the relative error of distance, so to reduce the speed error, time must be measured with much more precision.
Absolute mistake:
$$Er = \\frac{Δv}{v}$$
$$Δv = Er x v$$
$$Δv = 0.22 x 54.9$$
$$Δv = 12 cm/s$$
Therefore, the speed estimation with uncertainties is $$v ± Δv = (54.9 ± 12) m/s$$
When calculating the relative uncertainty, it is known which magnitude should be measured more precisely to reduce the total error of a derived magnitude.