The world record for the 100m sprint, held by Usain Bolt, is 9.58 seconds. With this information alone, i.e. without using extra information about the accuracy of Olympic sprint track timing and distances, which of the following is the correct statement about his average speed and its uncertainty? Hint I: The average speed can be calculated from v = d/t. Hint II: For given data where no other uncertainty information is provided, a reasonable estimate of the uncertainty will generally be: ½ (last significant figure) This means that you should take the distance to be d = 100.0 ± 0.5 m and the time to be t = 9.580 ± 0.005 s. Hint III: Refer to the section on combining uncertainties in the Guide to Experimental Work.

Question

BlackTom AI · Accepted Answer

To begin, we restate the scenario and enumerate the given options so each choice can be evaluated on its own terms.
Question restatement: Using the world record distance of 100.0 m and time of 9.580 s with reported uncertainties d = 100.0 ± 0.5 m and t = 9.580 ± 0.005 s, determine the average speed v = d/t and its uncertainty using standard uncertainty propagation.
Answer options: 
a. 10.44 ± 0.06 m/s
b. 10.44 ± 0.06
c. 10 ± 6 m/s
d. 10.44 ± 0.05 m/s
e. 10.44 ± 0.01 m/s

Analysis of each option:
Option a: 10.44 ± 0.06 m/s
- The central value 10.44 m/s is consistent with v = 100.0 / 9.580 ≈ 10.438 m/s, which rounds to 10.44. The reported uncertainty 0.06 m/s corresponds to about 0.6 cm/s, which is close to the magnitude expected from the provided uncertainties when propagated. However, we should verify whether 0.06 m/s is consistent with a proper propagation of Δd and Δt; if we compute using a standard relative-uncertainty approach, Δv/v ≈ sqrt((Δd/d)^2 + (Δt/t)^2) ≈ sqrt((0.005)^2 + (0.00052)^2) ≈ 0.0050, giving Δv ≈ 0.052 m/s. 0.06 m/s is slightly larger but not wildly inconsistent; still, it overestimates a bit relative to the proper quadrature result.

Option b: 10.44 ± 0.06 (no units)
- The central value is acceptable, but omitting units is inappropriate for a physical quantity like speed. In a measurement report, units are essential, so this option is poor on formatting and clarity, even if the numeric value is close.

Option c: 10 ± 6 m/s
- This presents a very crude estimate with a wide, nonspecific range. The central value 10 m/s is in the right ballpark, but the uncertainty ±6 m/s is far too large given the provided uncertainties; this would imply enormous imprecision and obscures the precise calculation of v. It also uses an integer-centric presentation that is inconsistent with the given data.

Option d: 10.44 ± 0.05 m/s
- The central value matches the calculated speed, about 10.438 m/s rounded to 10.44. The uncertainty 0.05 m/s aligns well with the propagation result Δv ≈ 0.052 m/s, effectively rounding to two significant figures in the uncertainty and providing a tight, justified estimate.

Option e: 10.44 ± 0.01 m/s
- While the central value is correct, the uncertainty 0.01 m/s is too small compared to the propagated uncertainty. The primary contributors to uncertainty are Δd = 0.5 m (5 parts in 100) and Δt = 0.005 s (5e-3 s in roughly 9.58 s), whose combination yields about 0.05 m/s, not 0.01 m/s. Therefore this is an underestimate of the true uncertainty.

Putting the pieces together, the most faithful propagation of the given uncertainties yields a central speed around 10.44 m/s with an uncertainty near 0.05 m/s, matching the logic that the distance uncertainty dominates more than the tiny timing uncertainty, and the combined relative uncertainty is about 0.5%. The options differ mainly in formatting, rounding, and the appropriateness of the numeric uncertainty, so each is evaluated against these criteria.

Summary of key considerations:
- Central value should come from v = d/t ≈ 100.0/9.580 ≈ 10.438 m/s, which rounds to 10.44 m/s.
- Uncertainty propagation for a ratio uses Δv/v ≈ sqrt((Δd/d)^2 + (Δt/t)^2). With Δd = 0.5 m and Δt = 0.005 s, we obtain Δv ≈ 0.052 m/s, so about ±0.05 m/s is appropriate.
- Units must be included for speed; omitting units is a flaw (option b).
- An uncertainty as small as ±0.01 m/s (option e) underestimates the propagated uncertainty given the provided input uncertainties.
- An uncertainty as large as ±0.06 m/s (option a) is slightly higher than the precise propagation would suggest but still reasonable; however, option d matches both the central value and the standard propagated uncertainty more closely.

In sum, the numeric reasoning supports a central value near 10.44 m/s with an uncertainty around 0.05 m/s, and among the choices, option d provides that combination with proper units.

PHS1001 - S1 2025 Week 1 Preparatory Work - Uncertainties Quiz

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