IB Physics SL S1.3 Mathematics Questions

Practise IB Physics SL S1.3 by selecting equations, manipulating units, reading graphs and checking mathematical reasonableness in physics problems.

Exam points

apply algebra, trigonometry, graph methods and proportional reasoning to physical relationships
use correct SI units, significant figures and uncertainty conventions in physical calculations
check mathematical results using dimensions, limiting cases, uncertainty and the physical context of the problem

Covered syllabus points

S1.3.1—General mathematics S1.3.2—Vectors and diagrams S1.3.3—Units and numerical values S1.3.4—Uncertainties S1.3.5—Graphing

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Study S1.3 Mathematics conceptsOpen the corresponding Concept lesson for explanations, examples and exam connections.

[Maximum number: 4]

A group of students is investigating refraction in a semi-circular glass block.

Light from a ray box enters the curved side of the block. The light passes through the block and leaves, refracted, at P .

(a)

The students vary the position of the ray box to obtain data to determine the refractive index of the glass. They use a protractor to collect values for the angles of incidence $\theta_{i}$ and refraction $\theta_{r}$ at P and record them on a table.

[ 1 ]

(i)

Complete the table.

[ 1 ]

(b)

They plot a graph of the variation with the sine of $\theta_{i}$ of the sine of $\theta_{r}$ .
They add uncertainty bars for $\sin \theta_{r}$ for the first and last data point and draw the best-fit line.

[ 3 ]

(i)

Determine the gradient of the students' best-fit line.

[ 2 ]

use two points from the line that use more than half the line of best fit
gradient =1.5

Marking guidance:

Allow (implicit) use of the origin.

Look for $\Delta \mathrm{x} \geq 0.3$

[2]

(ii)

Draw on the students' graph the line of maximum gradient.

[ 1 ]

[Maximum number: 3]

A glider is an aircraft with no engine. To be launched, a glider is uniformly accelerated from rest by a cable pulled by a motor that exerts a horizontal force on the glider throughout the launch.

At a particular instant in the flight the glider is losing 1.00 m of vertical height for every 6.00 m that it goes forward horizontally. At this instant, the horizontal speed of the glider is $12.5 \mathrm{~m} \mathrm{~s}^{-1}$ . Calculate the velocity of the glider. Give your answer to an appropriate number of significant figures.

Quote speed and direction together, to 3 significant figures. Accept a speed and direction together;
the accepted values are:
$\text{speed}=12.7\,\mathrm{m\,s^{-1}}$ $\text{direction}=9.46^{\circ}=0.165\,\mathrm{rad}\text{ below the horizontal}$ $\text{gradient}=-\frac{1}{6}$

[Maximum number: 6]

A group of students uses pressurized air to move a piston that forces a nail into a block of wood. A gauge is used to measure the pressure P of compressed air above atmospheric pressure. The nail enters the wood perpendicular to its surface.

The students use a ruler to measure the length of the nail which remains above the surface of the wood as shown. The depth of the nail inside the wood is d. All necessary length measurements are recorded using a ruler with uncertainty $\pm 1 \mathrm{~mm}$ .

(a)

The students systematically increase the pressure and calculate d.

[ 1 ]

(i)

State the absolute uncertainty of d.

[ 1 ]

«Addition of absolute uncertainties $\pm$ » 2 mm

Accept 0.002

(b)

The students suggest the following relationship between d and P :

d=k \sqrt{P}

where k is a constant.
To verify the relationship, the variation of d with $\sqrt{P}$ is plotted.
One data point is missing.

[ 5 ]

(i)

The percentage uncertainty in P is $\pm 8 \%$ . Determine the uncertainty for $\sqrt{P}$ when $d=33 \mathrm{~mm}$ and draw the uncertainty bar for this data point on the graph.

[ 3 ]

Alternative 1

«Uncertainty in $\sqrt{P}$ is» $\frac{1}{2} \times 0.08= \pm 4 \%$ $\pm 4 \% \times 500= \pm 20$

Uncertainty bar plotted correctly

Alternative 2
$\frac{0.08 \times 250000=20000 \quad }{\sqrt{250000+20000}-\sqrt{250000-20000}}= \pm 20$

Uncertainty bar plotted correctly

(ii)

The SI units of k can be expressed as $\mathrm{kg}^{-\frac{1}{2}} \mathrm{~m}^{x} \mathrm{~s}^{y}$ . Determine the values of x and y.

[ 2 ]

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