CAIE A-Level Physics A2: Paper 5 Planning & Analysis

[Maximum number: 3]

Carbon-15 $\left({ }_{6}^{15} \mathrm{C}\right)$ is an isotope of carbon that undergoes radioactive decay to nitrogen-15 $\left({ }_{7}^{15} \mathrm{~N}\right)$ , which is a stable isotope of nitrogen.

Radioactive decay is both a random and a spontaneous process.

(a)

A small sample of carbon-15 decays. The mass M of carbon-15 in the sample decreases with time t.

Fig. 10.1 shows the variation with t of the value of $\ln \left(M / 10^{-16} \mathrm{~g}\right)$ .

Fig. 10.1

[ 3 ]

(i)

Show that the decay constant $\lambda$ of carbon-15 is given by the magnitude of the gradient of your line in (b)(ii).

[ 1 ]

$M=M_{0} \exp (-\lambda t)$

so $\ln M=\ln M_{0}-\lambda t$ so gradient $=-\lambda($ and magnitude of gradient $=\lambda)$

(ii)

Use your line in (b)(ii) to determine $\lambda$ . Give a unit with your answer.

\begin{aligned} & \lambda= \\ & \text { unit } \end{aligned}

[ 2 ]

gradient =(-)(8.0-4.8) /(11.6-0) (allow any correct pair of values from Fig. 10.1)
$\lambda=0.28 \mathrm{~s}^{-1}$

[Maximum number: 10]

A student investigates the sound from a horn attached to a car, as shown in Fig. 2.1.

Fig. 2.1 (not to scale)

A microphone is placed at the side of the road and connected to a frequency meter. The car travels towards the microphone. The frequency f of the sound detected by the microphone is read from the frequency meter.

The speed of the car is measured by two speed detectors. The two measurements of speed are $v_{1}$ and $v_{2}$ . The average speed v of the car is determined from $v_{1}$ and $v_{2}$ .

The experiment is repeated for different speeds of the car.
It is suggested that f and v are related by the equation

f=\frac{f_{\mathrm{s}} k}{k-v}

where $f_{\mathrm{s}}$ is the frequency of the sound emitted by the horn and k is a constant.

(a)

A graph is plotted of $\frac{1}{f}$ on the y-axis against v on the x-axis.

Determine expressions for the gradient and y-intercept.

\begin{array}{r} \text { gradient }= \\ y \text {-intercept }= \end{array}

[ 1 ]

\text{gradient}=-\frac{1}{k f_s}

\text{y-intercept}=\frac{1}{f_s}

(b)

(c) (i) Plot a graph of $\frac{1}{f} / 10^{-3} \mathrm{~Hz}^{-1}$ against $v / \mathrm{m} \mathrm{s}^{-1}$ . Include error bars for v.

[ 2 ]

(c)

(ii) Draw the straight line of best fit and a worst acceptable straight line on your graph. Label both lines.
(iii) Determine the gradient of the line of best fit. Include the absolute uncertainty in your answer.
gradient =

Figure for Question (c) — CAIE A-Level Physics A2

[ 2 ]

(d)

Determine the y-intercept of the line of best fit. Include the absolute uncertainty in your answer.

y \text {-intercept }=

[ 2 ]

(c)(iii)

Gradient determined with clear substitution of data points into $\Delta y / \Delta x$ .

Gradient must be negative.

Distance between data points must be greater than half the length of the drawn line.

Gradient determined of worst acceptable line with clear substitution of data points into $\Delta y / \Delta x$ .

uncertainty = (gradient of line of best fit - gradient of worst acceptable line)

or

uncertainty =1 / 2 (steepest worst line gradient - shallowest worst line gradient)

2(c)(iv)

y-intercept determined by substitution of correct point with consistent power of ten in m and y into y=m x+c.

y-intercept of worst acceptable line determined by substitution into y=m x+c.

uncertainty =y-intercept of line of best fit -y-intercept of worst acceptable line

or

uncertainty =1 / 2 (steepest worst line y-intercept - shallowest worst line y-intercept)

Marking guidance:

Do not accept ECF from false origin method.

(e)

Using your answers to (a), (c)(iii) and (c)(iv), determine the values of $f_{\mathrm{s}}$ and k. Include appropriate units.

\begin{gathered} f_{\mathrm{s}}= \\ k= \end{gathered}

[ 2 ]

$f_{\mathrm{s}}$ determined using y-intercept and $f_{\mathrm{s}}$ given to 2 , 3 or 4 significant figures and k given to 2 or 3 significant figures.
$f_{s}=\frac{1}{y \text {-intercept }}$

k determined using gradient with method shown and $f_{\mathrm{s}}$ and k given with SI units with appropriate powers of ten.
$\begin{aligned} k=-\frac{y \text {-intercept }}{\text { gradient }} \text { or } k=-\frac{1}{\text { gradient } \times f_{s}} \text { Units of } f_{\mathrm{s}}: \mathrm{Hz} \text { Units of } k: \mathrm{m} \mathrm{~s}^{-1} \end{aligned}$

(f)

Determine the percentage uncertainty in k.

[ 1 ]

Percentage uncertainty in k with method shown.
$\text{percentage uncertainty}=\left(\frac{\Delta y\text{-intercept}}{y\text{-intercept}}+\frac{\Delta\text{ gradient}}{\text{gradient}}\right)\times100$
or correct substitution for max/min methods.

CAIE A-Level Physics A2 Paper 5 Planning Analysis and Evaluation Questions

Exam points

Covered syllabus points

Covered syllabus points

Question 1

Question (a)

Question (i)

Question (ii)

Question 2

Question (a)

Question (b)

Question (c)

Question (d)

Question (e)

Question (f)