Use of SI units and their prefixes: SI units derived.

SI units can be built from seven base units: kilogram, metre, second, ampere, kelvin, mole and candela. A derived unit is not a new basic quantity; it is a combination of these base units. To derive one, start from the defining equation for the quantity. That equation tells you which base units to substitute, collect and cancel. This skill supports unit conversion, dimensional consistency and checking whether an equation is physically possible. Use a consistent method. First, write the defining equation, for example speed equals distance divided by time. Second, replace each symbol with its SI base unit, not with a named derived unit. Third, collect repeated base units into powers, so metre times metre becomes metre squared. Fourth, cancel units that appear on both sides of a division. Finally, use negative indices for units in a denominator and compare your combination with the named SI unit. Let's apply the method to speed. Speed is distance divided by time, so its unit is metre divided by second, written m s⁻¹. Now consider force. Before I substitute, pause and ask which defining equation links force to mass and acceleration. The equation is F = ma. Substituting gives kilogram times metre per second squared, or kg m s⁻². This combination is the newton. Collect the powers and check the named unit. Energy follows from work, W = Fs. Using force in base units gives kg m s⁻² times metre, which simplifies to kg m² s⁻², the joule. Charge follows from Q = It, giving ampere times second, or A s, the coulomb. Notice that every factor is included and every repeated unit becomes a power. If a unit appears in a denominator, keep it as a negative index. These derived combinations are what you compare with named SI units. Three mistakes are common. First, substituting a named derived unit such as N into a defining equation. Expand N to kg m s⁻² first. Second, forgetting to square a unit when a quantity is squared. For kinetic energy, square every factor, giving kg m² s⁻². Third, treating the mole or kelvin as dimensionless. Include mol or K wherever the defining equation contains amount of substance or temperature. To derive an SI unit, write the defining equation, substitute base units, collect powers and cancel where possible. Keep negative indices for denominators, then compare the result with the named SI unit. This checks dimensional consistency and helps you judge whether an equation is physically possible. Before moving on, pause and choose one equation from your course, then derive its unit line by line.