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Physics · 01 / 12

Units: numbers have dimensions

Middle school · Worked example and 4 exercises with solutions, at your pace.

UnderstandSee an exampleGuided exercise3 progressive challenges

The principle, simply

A physical quantity combines a number and a unit. Convert before inserting values into a formula. Dimensions expose errors: velocity is length divided by time. Powers of units also change: 1 cm² = 10⁻⁴ m², not 10⁻² m².

v=100/5
THE PICTURE TO REMEMBERUnits: numbers have dimensions
  1. 01Quantity
  2. 02Consistent unit
  3. 03Conversion
  4. 04Dimension check

WORKED EXAMPLE

From principle to reasoning

100 m in 5 s gives 20 m/s. Multiply by 3.6 to convert to 72 km/h.

Solve a problem in this subject

  1. Draw the system

    Define the studied object, reference frame and positive axis. A force diagram shows external interactions acting on the system.

  2. Sort quantities

    Separate data and unknowns; convert units before substitution. 36 km/h = 10 m/s, so never insert 36 into a formula expecting SI speed.

  3. Write the model

    Choose a law matching assumptions: constant acceleration, thin lens, ohmic resistor. Isolate the unknown symbolically before rounding.

  4. Interpret

    Check dimensions and signs. Negative velocity gives direction on the chosen axis. Discuss scale and model limitations.

YOUR TURN

01 · Try before checking

Convert 72 km/h to m/s.

One hint

A kilometre is 1000 m; an hour is 3600 s.

Reveal and understand the solution

72 × 1000/3600 = 20 m/s. A second is a much shorter interval.

Compare method, conditions and answer; identify the exact part to revisit.

IMPROVE THROUGH PRACTICE

Three additional challenges with solutions

Start with the first challenge. Write your method before opening a hint; use the solution to find the first step that needs work.

Reflect after practising
02 · Recognise and understandLink to this exercise

Convert 72 km/h to m/s.

A hint to get started

Convert kilometres and hours separately.

Step-by-step solution
  1. 72 km is 72,000 m; one hour is 3,600 s.
  2. 72,000/3,600 = 20.
  3. The speed is 20 m/s.

Which step did you miss? Retry without help, then explain why the method works.

03 · Apply the methodLink to this exercise

Convert 250 cm² to m².

A hint to get started

Square the length conversion for an area.

Step-by-step solution
  1. 1 cm = 0.01 m, so 1 cm² = 0.0001 m².
  2. 250 × 0.0001 = 0.025.
  3. The area is 0.025 m², not 2.5 m².

Which step did you miss? Retry without help, then explain why the method works.

04 · Go furtherLink to this exercise

Can d = v + t represent distance?

A hint to get started

Only quantities with matching dimensions can be added.

Step-by-step solution
  1. v has dimension length/time, whereas t is time.
  2. Their sum does not have the dimension of length.
  3. For constant speed, d = v × t is dimensionally consistent.

Which step did you miss? Retry without help, then explain why the method works.

The trap to avoid

Do not mix minutes and seconds or convert an area as if it were a length.

Without notes, can you…

  • Explain the principle in your own words?
  • Rebuild the diagram and its connections?
  • Solve the exercise and justify the method?

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