考前速通

Key Relations: Gradient/Intercept to Constants

Linearized form	Gradient	y-intercept
$y = mx + c$	$m$	$c$
$\ln y = bx + \ln a$	$b$	$\ln a \Rightarrow a = e^{\,c}$
$\lg y = n \lg x + \lg a$	$n$	$\lg a \Rightarrow a = 10^{\,c}$
$y = a(1/x) + b$	$a$	$b$
$1/f = \frac{4}{v}L + \frac{4c}{v}$	$4/v \Rightarrow v = 4/m$	$4c/v \Rightarrow c = cv/4$

Constant from Gradient Workflow

read gradient m from graph
find formula: m = expression
rearrange: constant = f(m)
substitute numbers
add units

Constant from y-intercept Workflow

read intercept c from graph
find formula: c = expression
if ln/c -> constant = e^(c)
if lg/c -> constant = 10^(c)
otherwise constant = c
add units

Conclusion Template

"The theoretical value of $X = \_\_\_$ lies within / outside the experimental range $(X_{\text{exp}} \pm \Delta X) = (\_\_\_ \text{ to } \_\_\_)$. Therefore the results support / do not support the suggested relationship within experimental uncertainty."

Red Flags

信号	可能的问题
y-intercept from $\ln$ graph	必须取 $e^{\text{intercept}}$
y-intercept from $\lg$ graph	必须取 $10^{\text{intercept}}$
gradient 为负	检查常数是否应为正
题目给了已知常数	必须代入（如 $k = 1.38 \times 10^{-23}$）
答案没有单位	必扣分
理论值比较	必须提及 uncertainty range
扩展计算	用精确值，不要用四舍五入值

Quick Checklist

Part (d)(i): Calculate constant from gradient

• Correct expression linking gradient to constant
• Numerical substitution correct
• Units derived and stated

Part (d)(ii): Uncertainty in constant

• Error propagation formula correct
• All contributing terms included
• Answer: value $\pm$ uncertainty with units

Part (e): Extension calculation

• Correct formula used
• Previously calculated constant used correctly
• Final answer with unit

Part (f): Conclusion

• Experimental range stated (value $\pm$ uncertainty)
• Theoretical/expected value stated
• Compared correctly
• Clear "supports" or "does not support" with reasoning

提示

从 y-intercept 求常数时，务必判断是否需要取指数！$\ln a \rightarrow a = e^{\text{intercept}}$，$\lg a \rightarrow a = 10^{\text{intercept}}$。