Question

Starting position of an object is represented as \( x = 5.1 \pm 0.2 \text{ m} \) and the finishing position as \( y = 6.9 \pm 0.3 \text{ m} \). What will be the displacement and error in displacement?

• A. \( \text{Displacement = 1 m, Error = 0.5 m} \)
• B. \( \text{Displacement = 2 m, Error = 0.36 m} \)
• C. \( \text{Displacement = 1.8 m, Error = 0.36 m} \)
• D. \( \text{Displacement = 1.5 m, Error = 0.4 m} \)

Hint:

When subtracting or adding measurements with uncertainties, use the root-sum-square (RSS) method to find the total uncertainty:
\( \Delta Z = \sqrt{(\Delta A)^2 + (\Delta B)^2} \). This method assumes the uncertainties are independent and random, which is standard in experimental error analysis.

Answer 1

The Correct Option is C

To solve this problem, let's dive into the concept of displacement and error propagation in measurements.

1. What is Displacement?

Displacement is the change in position of an object. It is calculated as the difference between the final position and the initial position of the object:

\[ \text{Displacement} = y - x \] where: - \( x \) is the starting position, and - \( y \) is the finishing position.

2. What is Error in Displacement?

When dealing with measurements that have uncertainties (errors), the total error in a calculated quantity can be found by combining the individual errors of the measurements. In this case, we calculate the error in displacement by adding the absolute errors of the initial and final positions in quadrature (assuming the errors are independent):

\[ \text{Error in Displacement} = \sqrt{(\text{Error in } x)^2 + (\text{Error in } y)^2} \] where: - \(\text{Error in } x\) is the uncertainty in the initial position, and - \(\text{Error in } y\) is the uncertainty in the final position.

3. Given Values:

Starting position: \( x = 5.1 \pm 0.2 \, \text{m} \)
Finishing position: \( y = 6.9 \pm 0.3 \, \text{m} \)

4. Calculating the Displacement:

\[ \text{Displacement} = y - x = 6.9 \, \text{m} - 5.1 \, \text{m} = 1.8 \, \text{m} \]

5. Calculating the Error in Displacement:

\[ \text{Error in Displacement} = \sqrt{(0.2)^2 + (0.3)^2} = \sqrt{0.04 + 0.09} = \sqrt{0.13} \approx 0.36 \, \text{m} \]

6. Final Answer:

The displacement is 1.8 m, and the error in displacement is 0.36 m.