Distinguish scalar from vector quantities, use SI units and the metric prefixes, and convert measurements before substituting them into an equation (MA STE Introductory Physics, Motion and Forces).

What this topic is asking

Every physics problem starts with measured quantities, and the Massachusetts Introductory Physics MCAS expects you to handle them correctly before any equation appears. You need to tell a scalar from a vector, use SI units and the metric prefixes, and convert a measurement that is given in the wrong unit. These are not glamorous skills, but they are the difference between a right answer and a wrong one, because the reference-sheet formulas only work when the numbers are in consistent units.

Scalars and vectors

The distinction matters because direction changes the physics. Drive $300$ m east and then $300$ m back west, and the total distance traveled is $600$ m (a scalar, just adding up), but the displacement is zero (a vector, because you end where you started). The same split runs through the whole course:

A common MCAS task gives you a quantity and asks whether it is a scalar or a vector, or asks you to find a displacement when the motion reverses. The trick is always to ask: does direction change the answer? If yes, it is a vector.

SI units and the metric prefixes

Physics measurements are reported in SI units (the International System). The three you meet most are:

• length in meters (m)
• mass in kilograms (kg)
• time in seconds (s)

From these come the derived units: speed in meters per second (m/s), force in newtons (N, which is kg m/s squared), and energy in joules (J). The metric prefixes scale a unit up or down by powers of ten:

• kilo (k) means $\times 1000$, so $1$ km $= 1000$ m and $1$ kg $= 1000$ g.
• centi (c) means $\div 100$, so $1$ cm $= 0.01$ m.
• milli (m) means $\div 1000$, so $1$ mm $= 0.001$ m and $1$ ms $= 0.001$ s.

Because the prefixes are powers of ten, converting is multiplying or dividing by $10$, $100$, or $1000$, never anything awkward.

Why conversion comes first

The reference-sheet equations such as $v = \dfrac{d}{t}$ and $F = ma$ are written for SI units. If you put kilometers into a formula that expects meters, the number is wrong by a factor of a thousand. So the first step in many MCAS problems is to convert every quantity to meters, kilograms, and seconds.

The most common conversion in motion problems is speed from kilometers per hour to meters per second. Since $1$ km $= 1000$ m and $1$ hour $= 3600$ s:

The shortcut is to divide by $3.6$ to go from km/h to m/s, and multiply by $3.6$ to go back.

Reference-sheet note

The reference sheet lists the metric prefixes you may need and the formulas in SI units. It does not convert for you, so reading a value in grams, centimeters, or kilometers per hour and changing it to SI units is a skill you supply. Getting this wrong is one of the most common ways to lose an otherwise correct calculation.

Try this

Q1. State whether each is a scalar or a vector: mass, velocity, energy, force. [2]

• Cue. Mass scalar, velocity vector, energy scalar, force vector.

Q2. Convert $250$ cm to meters and $2.5$ kg to grams. [2]

• Cue. $250$ cm $= 2.5$ m (divide by $100$); $2.5$ kg $= 2500$ g (multiply by $1000$).