Students decide to measure a projectile's range for an initial projectile angle of 45°. This angle has many advantages, not the least being that since the expression for the range is proportional to the sine of twice this angle, errors in determining the angle do not contribute to errors in the range. As before, they make measurements to determine the initial speed. This time they find the initial speed to be 3.22 m/s with a relative uncertainty of 2.4%.
(a)What is the predicted range?
(b)What is the uncertainty in the predicted range? [Remember that you can treat the uncertainty in the sin(2) factor as zero since it contributes no errors at 45°.]
They launch the ball ten times and find the results for the range in centimeters to be 109.5, 109.6, 106.8, 112.5, 111.3, 106.5, 116.5, 110.1, 109.0, and 111.5.
(a)iv) What is the uncertainty in any single range measurement? [Hint: Think "standard deviation of measured values".]
(b)What is the standard error of the mean in the average range?
(c)Suppose the launch angle were the same for all ten trials, but only known (measured) to 1.3% . You would then expect the angular uncertainty to increase the spread in the measured range values by:
The average range is calculated to be 105.8
To find the predicted range, we need to use the formula for range:
Range = (initial speed ^ 2 * sin(2 * angle)) / gravitational acceleration
(a) Given that the initial speed is 3.22 m/s and the angle is 45°, we can calculate the predicted range using the formula. Since the angle is 45°, sin(2 * 45°) = sin(90°) = 1. Thus, the expression for the range simplifies to:
Range = (3.22^2 * 1) / gravitational acceleration
To calculate the predicted range, we need the value of the gravitational acceleration. Assuming it is 9.8 m/s^2, we can substitute this value into the equation:
Range = (3.22^2 * 1) / 9.8
(b) To determine the uncertainty in the predicted range, we need to consider the relative uncertainty in the initial speed. The relative uncertainty is given as 2.4%. Since the expression for the range is proportional to the sine of twice the angle, we can treat the uncertainty in the sin(2) factor as zero since it contributes no errors at 45°. Therefore, only the uncertainty in the initial speed affects the predicted range.
To find the uncertainty in the predicted range, we calculate the range for the maximum and minimum values of the initial speed within the relative uncertainty range. Let's call the maximum value (3.22 + 2.4%) and the minimum value (3.22 - 2.4%). Then, we calculate the range using these values and find the difference between the maximum and minimum range to determine the uncertainty.
(iv) To find the uncertainty in any single range measurement, we need to calculate the standard deviation of the measured range values. Given the range measurements of 109.5, 109.6, 106.8, 112.5, 111.3, 106.5, 116.5, 110.1, 109.0, and 111.5, we can calculate the standard deviation to obtain the uncertainty.
(b) To find the standard error of the mean in the average range, we divide the uncertainty in any single range measurement by the square root of the number of measurements (in this case, 10).
(c) If the launch angle is known (measured) to 1.3% for all ten trials, we expect the angular uncertainty to increase the spread in the measured range values by a factor of 1.3%. To determine the increase in spread, we can multiply the standard deviation of the measured range values by 1.3%.