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Mechanics is quantitative, so every quantity carries a unit and many carry a direction as well. Getting both right is not clerical detail — a units check will catch an algebraic error, and treating a vector as a scalar will produce an answer that is confidently wrong.
The big picture
Two habits from this lesson pay off across the whole strand. The first is that units can be treated algebraically: if you derive a formula and the units come out as anything other than the units of the quantity you wanted, the derivation is wrong. That check costs seconds and catches errors nothing else will. The second is the sign convention: choose a positive direction, write it down, and stick to it. Almost every "impossible" negative answer in mechanics is a direction being interpreted rather than a value being wrong.
What you'll be able to do
Mechanics uses three SI base units: the (m) for length, the (kg) for mass, and the (s) for time. Everything else is built from them.
A unit is a combination. Velocity is length per time, so . Acceleration is velocity per time, so . Force, from , is — which is given the name (N).
Note that mass is a base quantity but is a force, measured in newtons. A 5 kg object has a weight of about 49 N, and confusing the two is a persistent source of error.
Tip — Always convert to SI units before calculating. A speed in km/h or a mass in grams will silently produce an answer that is wrong by a factor of 1000 or 3.6.
Units behave algebraically, so both sides of a correct equation must reduce to the same combination. If they do not, the equation is wrong — no exceptions.
The check is quick and catches real mistakes. Take : the left is , and the right-hand term is . They agree.
Note what the check cannot do: dimensionless constants are invisible to it. and both pass, so a units check proves an equation is not obviously wrong rather than proving it right.
That last point is worth remembering: units verify the of a formula but never its numerical coefficients. A passing check narrows the possibilities without confirming the answer.
Scalars have size only: mass, time, distance, speed, energy, work, power. Vectors have size and direction: displacement, velocity, acceleration, force, weight, momentum.
The pairs that get confused are distance/displacement and speed/velocity. An object returning to its start has travelled a real distance but has zero displacement, and therefore zero average velocity despite a non-zero average speed.
Mechanics questions exploit this. "Find the distance travelled" and "find the displacement" have different answers whenever the direction of motion reverses.
Tip — Displacement, velocity and acceleration always need a direction stated — as a sign, a compass bearing, or in component form. A bare number is an incomplete answer for a vector.
In one-dimensional problems, direction is carried by a sign. Choose which direction is positive at the start, state it, and apply it consistently to every quantity in the problem.
The most common choice for vertical motion is upwards positive, which makes and an upward launch velocity positive. Taking downwards positive is equally valid; what is not valid is mixing the two.
A negative answer is information, not an error. A negative displacement means the object finished behind its starting point; a negative velocity means it is moving in the negative direction; a negative acceleration may mean slowing down or speeding up in the negative direction, depending on the velocity’s sign.
Note that "negative acceleration" does not mean "decelerating". It means the acceleration points in the negative direction — whether that speeds the object up or slows it down depends entirely on which way it is already moving.
Think like an examiner
Common misconceptions
Units and conventions
Stretch yourself
A student proposes that the maximum height reached by a projectile launched vertically at speed is . Use a units check to test this, then determine whether it is correct.
Hint — Check the units first, then derive the true expression using a suvat equation.
Questions students ask
Key takeaways
How this fits the course
Test yourself
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