You drop 12 balls off of a platform that is an unknown distance above the ground and measure the time it takes for the ball to fall to the ground. How far above the ground is the platform? Give a range for this answer that you can be 95% sure that the true distance lies within. Please assume that g = 9.80 m/s2. All times are given in seconds.
3.57
3.50
4.02
3.66
4.09
4.13
3.31
3.07
3.42
3.31
3.90
4.57
What is the distance above ground?
What is the uncertainty value in meters?
To find the distance above the ground, we can use the formula for the distance traveled by an object in free fall:
d = (1/2) * g * t^2
where d is the distance, g is the acceleration due to gravity (9.8 m/s^2), and t is the time.
1. First, calculate the average of the measured times:
t_avg = (3.57 + 3.50 + 4.02 + 3.66 + 4.09 + 4.13 + 3.31 + 3.07 + 3.42 + 3.31 + 3.90 + 4.57) / 12
t_avg = 42.95 / 12
t_avg = 3.58 s (rounded to two decimal places)
2. Next, calculate the standard deviation of the measured times. The standard deviation will give us an estimate of the uncertainty in our measurements.
- Calculate the deviation from the average for each measured time:
deviations = (3.57 - 3.58, 3.50 - 3.58, 4.02 - 3.58, 3.66 - 3.58, 4.09 - 3.58, 4.13 - 3.58, 3.31 - 3.58, 3.07 - 3.58,
3.42 - 3.58, 3.31 - 3.58, 3.90 - 3.58, 4.57 - 3.58)
deviations = (-0.01, -0.08, 0.44, 0.08, 0.51, 0.55, -0.27, -0.51, -0.16, -0.27, 0.32, 0.99)
- Calculate the sum of the squared deviations:
sum_sq_deviations = (-0.01)^2 + (-0.08)^2 + 0.44^2 + 0.08^2 + 0.51^2 + 0.55^2 + (-0.27)^2 + (-0.51)^2 + (-0.16)^2 + (-0.27)^2 + 0.32^2 + 0.99^2
sum_sq_deviations ≈ 2.1224
- Calculate the variance:
variance = sum_sq_deviations / (12 - 1)
variance = 2.1224 / 11
variance ≈ 0.192 °C (rounded to three decimal places)
- Calculate the standard deviation:
std_deviation = √variance
std_deviation ≈ √0.192
std_deviation ≈ 0.439 (rounded to three decimal places)
3. Now, calculate the uncertainty value in meters. Since the distance traveled is proportional to the square of the time, we can use the following relationship:
uncertainty = 2 * (std_deviation / t_avg) * d
We want to find the range within which we can be 95% sure the true distance lies. We'll use a t-distribution with 11 degrees of freedom and a confidence level of 95% to find the appropriate t-value.
t-value = 2.718
uncertainty = 2 * (0.439 / 3.58) * d
From the t-distribution table, we know that the area under the curve for a t-value of 2.718 is 0.95. This means that the range we're looking for is within ±2.718 standard deviations.
4. Rearranging the uncertainty equation, we can solve for d:
d = (uncertainty * t_avg) / (2 * std_deviation)
Plugging in the values, we have:
d = (2.718 * 3.58) / (2 * 0.439)
d ≈ 7.319 m (rounded to three decimal places)
Therefore, the distance above the ground is approximately 7.319 meters.
To find the uncertainty value in meters, we can substitute the calculated d value to the uncertainty equation:
uncertainty = 2 * (0.439 / 3.58) * 7.319
uncertainty ≈ 0.901 m (rounded to three decimal places)
So, the uncertainty value for the distance above the ground is approximately 0.901 meters.