Practice Dimensional Reasoning in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

Using the units and dimensions of physical quantities to check formulas, guide derivations, and eliminate impossible answers.

Units must balance on both sides of any physical equation — if the units do not match, the formula is wrong regardless of the numbers.

Showing a random 20 of 50 problems.

Example 1

medium
Use dimensional reasoning to find the units of impulse J=F ΔtJ = F\,\Delta t, given FF in N, tt in s.

Example 2

easy
A force has units kg·m/s2^2. Its name is ____.

Example 3

hard
Express the SI base-unit form of joules (J), watts (W), and volts (V).

Example 4

easy
Convert 60 km/h to m/s using dimensional analysis.

Example 5

medium
A student writes force=mv\text{force} = mv instead of mama. Which units mismatch reveals the error?

Example 6

medium
A student proposes kinetic energy KE = m*v^2 (no 1/2). Can dimensional reasoning detect the missing 1/2 factor? Explain.

Example 7

medium
A pendulum period is claimed to be T = 2*pi*sqrt(L/g). Verify the units give seconds, with L in m and g in m/s^2.

Example 8

medium
A rate is given as 90 km/h. Convert it to m/s by carrying units.

Example 9

challenge
A 'BMI' is defined as mass (kg) divided by height² (m²). Why is the resulting unit kg/m² acceptable even though it's not a familiar named unit?

Example 10

hard
Convert 9.8 m/s² to ft/s². (1 m ≈ 3.281 ft.)

Example 11

hard
Use dimensional analysis to guess how the speed cc of waves on deep water depends on gravitational acceleration gg and wavelength λ\lambda.

Example 12

easy
What are the units of frequency, the reciprocal of period in seconds?

Example 13

easy
Convert 30 minutes to seconds using unit cancellation.

Example 14

medium
A student writes E=mcE = mc for Einstein's mass-energy formula. Use dimensional reasoning to explain why this cannot be correct and what the correct formula should be.

Example 15

hard
A formula claims r=vt2/2r = vt^2/2 for distance traveled. Use units to determine whether vv should be velocity or acceleration.

Example 16

hard
Why can't dimensional reasoning alone determine whether sin⁡θ\sin\theta or cos⁡θ\cos\theta appears in a projectile range formula?

Example 17

easy
A formula gives time t = sqrt(2h/g), with h in meters and g in m/s^2. Check: what are the units inside the square root?

Example 18

easy
Convert 5 ft to inches (1 ft = 12 in).

Example 19

medium
Why does ln⁡(x)\ln(x) require xx to be dimensionless?

Example 20

medium
A quantity Q satisfies Q = a/t^2 and has units of m/s^2. If t is in seconds, what are the units of a?
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MeasurementScaling Laws