A beam of protons is directed from the left onto a small target. The interactions produce...
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A beam of protons is directed from the left onto a small target. The interactions produce secondary particles, which travel to the right after emitting from the target. The y coordinates of the secondary particles are measured by a series of detector planes that are accurately situated at x = 10, 14, 18, 22, 26, 30 cm downstream from the target (assume that the target is at x = 0 cm). The detectors at the first and the last planes have an accuracy of ±0.10 cm for the y measurements, while that for the other four planes is ±0.30 cm. The following y coordinates are obtained for the six planes after one specific interaction: 2.02, 2.26, 3.24, 3.33, 3.93, 4.03 cm. We want to determine whether these measurements are consistent with the hypothesis that the particle traces a straight track given by y = a + bx. a) Perform a least squares straight line fit to the data: find the weighted mean x position(x), the intercept a, the slope b, and the height a' at x =(x). Determine Smin and the number of degrees of freedom v. b) Decide whether this track is consistent with coming from the target (situated at x = 0 cm) and extends from y = -1 cm to y = +1 cm, as follows: Determine the accuracies with which a' and b are determined, and then use the relation y = a + bx' (where x¹ = x -(x)) to calculate the error & on the extrapolated position yo of the track at the x' value corresponding to x = 0. Then decide whether, given yo and E, the track is consistent with being produced in the target. c) How would your answers to part (b) have changed if the accuracies of determining the y positions were ten times smaller, i.e., 10.01 cm and ±0.03 cm? A beam of protons is directed from the left onto a small target. The interactions produce secondary particles, which travel to the right after emitting from the target. The y coordinates of the secondary particles are measured by a series of detector planes that are accurately situated at x = 10, 14, 18, 22, 26, 30 cm downstream from the target (assume that the target is at x = 0 cm). The detectors at the first and the last planes have an accuracy of ±0.10 cm for the y measurements, while that for the other four planes is ±0.30 cm. The following y coordinates are obtained for the six planes after one specific interaction: 2.02, 2.26, 3.24, 3.33, 3.93, 4.03 cm. We want to determine whether these measurements are consistent with the hypothesis that the particle traces a straight track given by y = a + bx. a) Perform a least squares straight line fit to the data: find the weighted mean x position(x), the intercept a, the slope b, and the height a' at x =(x). Determine Smin and the number of degrees of freedom v. b) Decide whether this track is consistent with coming from the target (situated at x = 0 cm) and extends from y = -1 cm to y = +1 cm, as follows: Determine the accuracies with which a' and b are determined, and then use the relation y = a + bx' (where x¹ = x -(x)) to calculate the error & on the extrapolated position yo of the track at the x' value corresponding to x = 0. Then decide whether, given yo and E, the track is consistent with being produced in the target. c) How would your answers to part (b) have changed if the accuracies of determining the y positions were ten times smaller, i.e., 10.01 cm and ±0.03 cm?
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a Perform a least squares straight line fit to the data find the weighted mean x positionx the intercept a the slope b and the height a at x x Determine Smin and the number of degrees of freedom v The ... View the full answer
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