IB Physics SL S1.3 Mathematics Questions

SLSL HLHL

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

156 questions in this syllabus scope, with question parts counted together.

Exam points & syllabus

Exam points

Identify the requested ratio and relevant length, area or volume scaling before cancelling powers or estimating an order of magnitude.
Manipulate the expression actually supplied: isolate a logarithmic ratio, use consistent angle mode and match coefficients before substituting values.
Build a rate from all supplied factors and units; convert elapsed times and use the stated bit/byte definition for storage calculations.
Resolve perpendicular vector components with geometry, and give both magnitude and direction when reporting a velocity.
Use vector direction as well as magnitude: reverse the subtracted vector, apply resultant bounds and cancel unwanted velocity components.
Distinguish scalars from vectors and resolve forces relative to the relevant surface or axes in the given diagram.
Distinguish SI base units from derived units, and expand the latter fully when fundamental units are requested.
Rearrange a supplied equation for a constant and substitute units into every factor, including powers and roots, before simplifying.
Convert prefixes to a common unit before comparing values, and distinguish significant figures from instrument decimal-place precision.
For sums or differences add absolute uncertainty bounds; for products or quotients add fractional bounds, weighted by each power’s magnitude.
Convert a combined fractional uncertainty to an absolute uncertainty using the result, then round the value and uncertainty to compatible precision.
Read an uncertainty bar’s half-span using the actual axis scale, and distinguish a small absolute error from a small relative error.
Use a justified half-range for repeated measurements, and scale an uncertainty when dividing a measured total across repeated intervals.
Choose a change that reduces the relevant fractional uncertainty: increase the measured change or interval, or improve its measurement precision.
Match transformed axes to the supplied model before identifying a slope or intercept; a parameter may be a root or power of the slope.
Use well-separated points on the fitted line for a gradient, showing both coordinate changes and all scale factors.
Draw one clear best-fit line or smooth curve consistent with the data and its bars, without point-to-point joins or an unjustified forced origin.
Distinguish linearity from direct proportionality: the latter requires a straight line through the origin, while a meaningful intercept may be nonzero.
Constrain extreme-gradient lines by the uncertainty bars and use their permitted intercept spread when estimating intercept uncertainty.
Use a tangent at the specified point for an instantaneous rate; an arbitrary chord or later tangent answers a different question.
Determine the meaning of an area from its axes; use velocity change, distance or height only where the given graph supports that interpretation.
Transform source measurements before plotting missing points and read extrema, fractional levels or extrapolated values from the stated curve.
When converting an echo time-versus-length gradient to sound speed, include the out-and-back path and the graph’s length/time units.

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

Exam points

Apply geometric scaling to the relevant dimension: doubling a radius quadruples area; a volume ratio needs a cube root to become a length ratio.
Divide powers of ten in the stated ratio direction; compare the resulting exponents to find a difference in orders of magnitude.
Check an estimate against a familiar physical scale, such as an apple mass of order 10⁻¹ kg.
Build a data rate from samples per second and information per sample, converting bits and bytes only with the supplied definition.
Determine a rate by dividing the total change by elapsed time, with consistent time and area units.
Use a right-triangle ratio to find a sloping path, then subtract the direct path; do not treat the sloping length as the extra distance.
For a supplied base-ten expression 10 log(P/P₀)=k, isolate the ratio as P/P₀=10^(k/10) before calculating P.
Use magnification as the image-to-object length ratio and keep both lengths in consistent units.
For equal square pixels, divide total area by pixel count and take the square root to find a side; reverse the area ratio to find the count.
For a show-that volume, show the full radius substitution or a result with enough significant figures; use the geometric volume of the named object.
Use calculator degree mode for angles given in degrees, and report the requested significant figures in trigonometric and logarithmic tables.
Match coefficients in supplied equations, convert the stated mass to kilograms, and derive a constant’s SI unit from the whole equation.

[Maximum number: 1]

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 .

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.

Complete the table.

Exam points

Identify vector quantities by their direction as well as magnitude; power, energy, mass and temperature are scalars.
For perpendicular components, use the sum of their squares for the resultant magnitude; rearrange it to find a missing component.
Report a velocity with both speed and direction, including whether it is above or below the stated reference and the required precision.
Resolve weight normal and parallel to an incline; for the pictured block at rest, N=W cos θ and friction balances W sin θ.
Construct a vector difference by adding the reversed second vector; check the resultant direction against the labelled answer arrows.
For two forces of magnitudes F₁ and F₂, a possible resultant magnitude lies between |F₁−F₂| and F₁+F₂, including both limits.
For straight motion across a flowing river, steer with a component opposing the current so the unwanted ground-velocity component cancels.

[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:
$\text{speed}=12.7\,\mathrm{m\,s^{-1}}$ $\text{direction}=9.46^{\circ}=0.165\,\mathrm{rad}\text{ below the horizontal}$ $\text{gradient}=-\frac{1}{6}$

Exam points

Distinguish SI base units from derived units: ampere is a base unit, while coulomb equals ampere second.
Use momentum units kg m s⁻¹, energy units kg m² s⁻² and power units kg m² s⁻³; keep the time exponents distinct.
Write pressure as kg m⁻¹ s⁻² and expand potential difference to kg m² s⁻³ A⁻¹ when fundamental units are requested.
Derive intensity from power per area; the metre-squared factors cancel to give kg s⁻³.
For a dimensional consistency proof, show the units of each quantity, substitute them into the equation and simplify to the stated result.
In a sum, every term has the same dimensions; divide by the multiplying variable to find the units of a coefficient.
Rearrange the supplied equation for the constant before replacing each quantity by its units; apply powers and roots to units too.
Find gradient units by dividing vertical-axis units by horizontal-axis units, including powers shown on the axes.
Keep heat capacity, specific heat capacity and heat capacity per area distinct; cancel shared factors when forming ratios of units.
Recognize pressure times volume as an energy unit, such as Pa m³, N m or J.
Convert joules to electronvolts before applying an energy prefix, and convert all prefixed lengths to one common unit before comparing them.
Use supplied bit/byte and storage prefixes consistently; round down a capacity quotient when only complete stored pictures count.
Leading zeros do not count as significant figures, but a final zero after the decimal point can; 0.0450 has three significant figures.
Relate recorded decimal places to the instrument’s fixed precision; the number of significant figures can change with the reading’s size.
For dimensional constants, distinguish a magnetic-field unit T (tesla) from a variable named T; use the meaning defined in the question.
Express μ₀ε₀ in base units by substituting C=A s and cancelling the remaining force and current units, giving m⁻² s².
Distinguish distance units from angular units; metres and radians form one valid pair for the stated parallax geometry.
For the field-strength question, state N kg⁻¹ explicitly; its scheme rejects a bare m s⁻² despite dimensional equivalence.
A percentage used as a dimensionless variable adds no physical unit; the fitted coefficient retains the required temperature-change unit.

[Maximum number: 2]

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 $\pm 1 \mathrm{~mm}$ .

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.

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.

Exam points

Distinguish absolute, fractional and percentage uncertainty: divide the absolute uncertainty by the measured value, then multiply by 100 for percent.
For a sum or difference, add absolute uncertainty bounds; apply any exact numerical multiplier to the resulting absolute uncertainty.
For products and quotients, add fractional or percentage uncertainties; division does not mean subtracting uncertainty contributions.
For a power xⁿ, multiply the fractional uncertainty by |n|; square roots and cube roots reduce the fractional contribution by their power.
Exact constants such as π do not add fractional uncertainty; quantities with the same power of a shared measured length have equal fractional uncertainty.
Convert the combined fractional uncertainty back to an absolute uncertainty by multiplying by the calculated result.
Round the uncertainty to the requested significant figures and round the central result to the corresponding decimal place.
Read an error bar’s half-span as the absolute uncertainty; check the axis scale and units before calculating a fraction or percentage.
Draw uncertainty bars symmetrically about the point, in the direction of the uncertain variable and with the calculated half-width.
Scale an absolute uncertainty when dividing a measured total time by its number of oscillations to obtain one period.
Timing many oscillations can reduce the fractional timing uncertainty when the start-and-stop absolute uncertainty stays similar.
Use the task’s range-based estimate for repeated readings, taking half the relevant range and showing any justified anomaly choice.
With equal absolute uncertainty, a larger measured value has a smaller fractional uncertainty; keep this distinct from changing the instrument.
A difference between nearly equal measurements can have a large fractional uncertainty even when each measurement has a small absolute uncertainty.
Correcting a zero offset does not make its measurement exact; include both reading uncertainties when the corrected value is a measured difference.
In resistivity or density calculations, apply the diameter’s power to its fractional uncertainty rather than treating diameter as a first-power factor.
To reduce a spacing’s absolute uncertainty, measure across several intervals and divide by the number of intervals, not the number of maxima.
Reduce fractional uncertainty by increasing the relevant measured change or improving precision, using a modification appropriate to the given setup.
A constant percentage uncertainty implies a changing absolute uncertainty; check whether that assumption fits the instruments and procedure.
If an uncertainty contribution is negligible, state why before omitting it; a missing printed uncertainty does not generally prove it is zero.
Follow the coordinate and model actually supplied for an empirical relation; changing a temperature origin can change a fitted power-law form.

[Maximum number: 4]

(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.

[ 3 ]

(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» $\frac12\times0.08=\pm4\%$

\pm4\%\times500=\pm20

Uncertainty bar plotted correctly

Alternative 2

0.08\times250000=20000

\frac{\sqrt{250000+20000}-\sqrt{250000-20000}}{2}=\pm20

Uncertainty bar plotted correctly

Allow ECF for MP2 and MP3
For MP3 apply Uncertainty tolerance of ±5 on each end of plotted error bar

Exam points

Match the plotted axes to y=mx+c before identifying a gradient or intercept; the physical constant may be a power or root of the gradient.
For a gradient, choose well-separated points on the fitted line, show the change in each coordinate and include all axis scale factors.
Draw one clear best-fit line or smooth curve that follows the data and its uncertainty bars; avoid point-to-point joins and doubled lines.
Use the supplied model to choose a linearizing transformation, then identify the resulting slope and intercept algebraically.
Distinguish a straight relationship from direct proportionality: the latter requires a straight line through the origin.
Do not force a best-fit line through the origin when the task or model does not require it; a nonzero intercept may be meaningful.
Convert the original data into the plotted variables and units before locating a missing point, including reciprocals, squares or cubes.
Draw uncertainty bars about the correct point using the absolute half-width; convert a given percentage into axis units first.
An uncertainty bar can be too small to show on the chosen scale; that does not make the measurement uncertainty zero.
An extreme-gradient line must remain consistent with the uncertainty bars; being steeper than the best-fit line alone is insufficient.
Find intercept uncertainty from the allowed extreme lines, then report the intercept and uncertainty with compatible precision.
To obtain a local rate from a curve, draw a tangent at the specified point and calculate its gradient using a sufficiently large triangle.
Use the initial tangent when the requested rate is initial; a gradient between arbitrary later curve points does not represent that rate.
Check the physical meaning of an area from its axes: speed–time gives distance and acceleration–time gives velocity change.
For maximum height from a velocity graph, use the signed area up to the first zero velocity, excluding later descent or bounce.
Relate a displacement graph’s gradient to velocity and a velocity graph’s changing gradient to acceleration when drawing qualitative motion graphs.
Read a required maximum, minimum or fractional level from the stated curve, checking its actual starting value and axis scales.
Use the model and graph together to extract a parameter; do not substitute a raw point where the task requires a fitted gradient.
Extrapolate a suitable fitted curve to the requested axis when estimating an initial value; the first measured value can occur later.
For an echo experiment, account for the out-and-back path when connecting a time-versus-length gradient to sound speed.
Use supplied circuit connections with an I–V graph: components in series carry the same current, so branch currents must not be added.
Test a supplied exponential hypothesis with several readings, checking repeated halving intervals or constant ratios at equal time intervals.
Use a particle’s oscillation and phase when plotting its later position on a wave diagram; do not move the particle along with the travelling wave.

[Maximum number: 1]

A cylindrical cork of height H and cross-sectional area A is floating stationary in water. Its depth below the water surface is D.

Identify the temperature at which water has its maximum density.

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