题型分析

Question Type 1: Uncertainty in Derived Quantities (Table)

如何识别

Part (b) — "Calculate and record values of $Y$. Include the absolute uncertainties."

标准解题方法

1. Calculate the derived quantity for each row
2. Use the uncertainty propagation formula
3. Record uncertainty beside each value
4. Use consistent significant figures

Example 1 — 9702/s20/qp/51 Q2(b)

Values of $t$ and $V$ given. $\Delta V = 0.2$ V for all. Calculate $\ln(V/\text{V})$ with uncertainties.

MS 展开查看

MS:

• M1: $\ln(V/\text{V})$ values calculated correctly
• A1: $\Delta(\ln V) = \Delta V / V$ producing uncertainties from $\pm 0.03$ to $\pm 0.14$

Example 2 — 9702/w20/qp/52 Q2(b)

Values of $d$ given with $\Delta d$. Calculate $1/d$ with uncertainties.

MS 展开查看

MS:

• M1: $1/d$ values correct
• A1: $\Delta(1/d) = \Delta d / d^2$ producing decreasing uncertainties

Example 3 — 9702/s22/qp/51 Q2(b)

Values of $M$ given. Calculate $\lg(M/10^{30}\ \text{kg})$ with uncertainties.

MS 展开查看

MS:

• M1: $\lg(M/10^{30})$ values correct
• A1: $\Delta(\lg M) = 0.434 \times \Delta M / M$

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Question Type 2: Gradient Uncertainty

如何识别

Part (c)(iii) — "Determine gradient. Include absolute uncertainty."

标准解题方法

1. Calculate gradient of best fit line (large triangle)
2. Calculate gradient of worst acceptable line
3. Uncertainty = |best - worst|
4. Answer: $\text{gradient} \pm \text{uncertainty}$

Example 1 — 9702/s20/qp/51 Q2(c)(iii)

From graph of $\ln V$ against $t$, determine gradient $k$ with its absolute uncertainty.

MS 展开查看

MS:

• M1: Best fit gradient from large triangle, $\Delta\ln V / \Delta t$
• A1: Worst acceptable gradient through error bars
• A1: $\Delta k = |k_{\text{best}} - k_{\text{worst}}|$

Example 2 — 9702/w22/qp/52 Q2(c)(iii)

From graph of $V$ against $1/d$, determine gradient with uncertainty.

MS 展开查看

MS:

• M1: Best fit line gradient using large triangle method
• A1: Worst acceptable line through all error bars
• A1: Uncertainty = absolute difference

Example 3 — 9702/s23/qp/51 Q2(c)(iii)

From graph of $\lg I$ against $\lg V$, determine $n$ (gradient) with uncertainty.

MS 展开查看

MS:

• M1: Best fit gradient = $n$
• A1: Worst fit gradient calculated
• A1: $\Delta n = |n_{\text{best}} - n_{\text{worst}}|$ given to 1 s.f.

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Question Type 3: Combined Uncertainty in Final Constant

如何识别

Part (d)(ii) — "Determine the absolute/percentage uncertainty in constant $X$."

标准解题方法

1. If $C = -\frac{1}{R \times \text{gradient}}$, then:
2. $\frac{\Delta C}{C} = \frac{\Delta R}{R} + \frac{\Delta \text{gradient}}{\text{gradient}}$
3. $\Delta C = C \times \left(\frac{\Delta R}{R} + \frac{\Delta \text{gradient}}{\text{gradient}}\right)$

Example 1 — 9702/s22/qp/51 Q2(d)(ii)

Given $C = -\frac{1}{R \times k}$, find percentage uncertainty in $C$ using $\Delta R$ and $\Delta k$.

MS 展开查看

MS:

• M1: $\frac{\Delta C}{C} = \frac{\Delta R}{R} + \frac{\Delta k}{k}$
• A1: Percentage uncertainty in $C$ calculated correctly
• A1: Final $C$ stated as value $\pm$ absolute uncertainty

Example 2 — 9702/w20/qp/52 Q2(d)(ii)

Using $R = \frac{\text{gradient}}{T}$ and known $\Delta T$, find uncertainty in $R$.

MS 展开查看

MS:

• M1: $\frac{\Delta R}{R} = \frac{\Delta \text{gradient}}{\text{gradient}} + \frac{\Delta T}{T}$
• A1: Correct $\Delta R$ calculated
• A1: Answer with consistent units

Example 3 — 9702/s23/qp/51 Q2(d)(ii)

From $A = \frac{k}{10^{\text{intercept}}}$, find percentage uncertainty in $A$ given $\Delta k$ and $\Delta \text{intercept}$.

MS 展开查看

MS:

• M1: $\frac{\Delta A}{A} = \frac{\Delta k}{k} + \ln 10 \times \Delta(\text{intercept})$
• A1: Correct propagation applied
• A1: Absolute uncertainty in $A$ to 1 s.f.

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Question Type 4: y-intercept Uncertainty

如何识别

Part (c)(iv) — "Determine y-intercept. Include absolute uncertainty."

标准解题方法

1. Read y-intercept from best fit line
2. Read y-intercept from worst acceptable line
3. Uncertainty = $|$best intercept $-$ worst intercept$|$

Example 1 — 9702/w22/qp/52 Q2(c)(iv)

Determine y-intercept $\ln V_0$ with its uncertainty.

MS 展开查看

MS:

• M1: Best fit y-intercept read correctly
• A1: Worst fit y-intercept read correctly
• A1: $\Delta(\text{intercept}) = |\text{best} - \text{worst}|$

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常见陷阱

Uncertainty 陷阱

• 表格中 uncertainty 的有效数字位数应与原始数据匹配（通常 1-2 s.f.）
• worst line 必须通过所有 error bars，而不是随便画
• 组合误差时误用加法/乘法规则
• 忘记转换 absolute 和 percentage uncertainty