Scalars and vectors
A scalar is fully described by a number and a unit: a mass of 3 kg, a time of 12 s, a temperature of 20 degrees Celsius. A vector also needs a direction: a velocity of 12 m/s north, a force of 40 N downward, a displacement of 5 m east. Displacement is the straight arrow from the start to the end of a trip; distance travelled is a scalar and is the length of the actual path. Speed is a scalar, velocity is a vector.
Units, prefixes and conversions
Physics uses the SI system: metres, kilograms and seconds are the base units for length, mass and time. Prefixes scale them: kilo means 1000, centi means one hundredth, milli means one thousandth. To convert, multiply by a fraction equal to 1 so that the unwanted unit cancels. Since 1 km is 1000 m and 1 h is 3600 s, 1 km/h equals 1/3.6 m/s.
Careful with squares and cubes: 1 cm is 0.01 m, so 1 cm³ is (0.01)³ = 10⁻⁶ m³, not 10⁻² m³.
Significant figures
Count digits from the first non-zero digit onward, and keep trailing zeros that follow a decimal point. In 0.0030 the leading zeros do not count, so there are two significant figures. When you multiply or divide, round the answer to the fewest significant figures among the inputs. When you add or subtract, round to the fewest decimal places.
Dimensional analysis
Every term in a correct equation has the same dimensions. A term like v t (speed times time) is a length, and so is a t² (acceleration times time squared), so d = v t + ½ a t² can be right, while v + a t² mixes a speed with a length and cannot. This is a quick sanity check before you trust a formula.
Adding vectors
Resolve each vector into components along perpendicular axes: for a magnitude A at an angle θ above the x-axis, A_x = A cos θ and A_y = A sin θ. Add the x-components together and the y-components together. The resultant has magnitude R = √(R_x² + R_y²) and direction θ = tan⁻¹(R_y / R_x), measured from the x-axis. Take care with signs: west and south are negative if east and north are positive. To subtract, add the opposite vector.
Worked example
A hiker walks 9.0 km east and then 12.0 km north. Find the net displacement.
- The components are R_x = 9.0 km and R_y = 12.0 km.
- Magnitude: √(9.0² + 12.0²) = √225 = 15.0 km.
- Direction: tan⁻¹(12.0 / 9.0) = 53.1 degrees north of east.
The distance walked is 21.0 km, but the displacement is only 15.0 km, because displacement is a straight-line vector.
A second case: a 40 m/s velocity at 25 degrees above the horizontal has horizontal component 40 cos 25 = 36.3 m/s and vertical component 40 sin 25 = 16.9 m/s.
Common mistakes
- Adding magnitudes of vectors that are not in the same direction.
- Swapping sine and cosine; check which side of the triangle is next to the angle.
- Forgetting the square root in the magnitude formula.
- Reporting every digit on the calculator.