For an experiment in this unit, you measure the time of a ball falling from rest as 3.4 ± 0.2 s and you want to calculate the distance the ball has fallen during that time, where you are taking the acceleration due to gravity as a = 9.8 ± 0.1 ms-2. What is the absolute uncertainty (to 1 sig fig) in the distance, calculated from the given values?
Question
For an experiment in this unit, you measure the time of a ball falling from rest as 3.4 ± 0.2 s and you want to calculate the distance the ball has fallen during that time, where you are taking the acceleration due to gravity as a = 9.8 ± 0.1 ms-2. What is the absolute uncertainty (to 1 sig fig) in the distance, calculated from the given values?
Solution
To calculate the absolute uncertainty in the distance, we first need to calculate the distance itself using the formula for distance fallen under constant acceleration, which is d = 0.5 * a * t^2.
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Calculate the distance: d = 0.5 * 9.8 m/s^2 * (3.4 s)^2 = 55.652 m
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Calculate the fractional uncertainties for both time and acceleration: For time: Δt/t = 0.2 s / 3.4 s = 0.0588 For acceleration: Δa/a = 0.1 m/s^2 / 9.8 m/s^2 = 0.0102
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Since the time is squared in the formula, we double the fractional uncertainty for time: 2 * 0.0588 = 0.1176
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Add the fractional uncertainties for time and acceleration: 0.1176 + 0.0102 = 0.1278
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Multiply the total fractional uncertainty by the distance to get the absolute uncertainty: Δd = d * 0.1278 = 55.652 m * 0.1278 = 7.1 m
So, the absolute uncertainty in the distance is 7.1 m.
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