37 exam-style questions on Mathematics — topic T3 of Skills in the study of physics in the IB Physics syllabus — with answers and explanations. Samples below; the full set is free with an account.
A student writes the kinetic energy of a moving mass as E = ½mv. A check of units shows the equation must be wrong. Why?
Answer: mv has units kg m s⁻¹, but energy needs kg m² s⁻²
The units on each side of a correct equation must match. mv gives kg m s⁻¹ (momentum), not the joule's kg m² s⁻²; the correct form is E = ½mv². Dimensionless factors like ½ do not affect units.
The period of a simple pendulum is given by T = 2π√(l/g). Which pair of quantities gives a straight line through the origin when plotted against each other?
Answer: T² on the vertical axis against l on the horizontal axis
Squaring gives T² = (4π²/g)·l, which has the form y = mx. Plotting T against l gives a curve, so the gradient has no simple meaning.
Calculate the product (4.0 × 10⁵) × (2.0 × 10⁻³).
Answer: 8.0 × 10²
Multiply the mantissas (4.0 × 2.0 = 8.0) and add the exponents (5 + (−3) = 2). Subtracting the exponents instead of adding them gives 8.0 × 10⁸.
An object starts from rest with uniform acceleration a, so that s = ½at². A graph of s against t² is a straight line through the origin. What does its gradient represent?
Answer: Half of the acceleration, that is a/2
Comparing s = ½at² with y = mx gives m = ½a, so the acceleration is twice the gradient. Reading the gradient straight off as a is the standard error in linearised graphs.
In the relationship F = kx², F is measured in newtons and x in metres. What is the SI unit of the constant k?
Answer: N m⁻²
Rearranging gives k = F/x², so the unit is N / m² = N m⁻². Homogeneity of units is the quickest check that a rearrangement has been done correctly.
The corrected count rate C of a radioactive source falls exponentially with time t. Which plot of the data gives a straight line?
Answer: ln C against t
Taking natural logarithms of an exponential fall gives ln C = ln C₀ − kt, which has the form y = mx + c with ln C on the vertical axis and t on the horizontal one. Taking the logarithm of the time as well straightens power laws, not exponentials.
That was 6 of 37 Mathematics questions.
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