Units, Measurement and Motion

Measurement and SI units

To measure is to compare a quantity with a standard unit. The world uses the SI system (Systeme International).

These are the seven base units. Others are derived: area (m^2), volume (m^3), speed (m/s), density (kg/m^3), force (newton), pressure (pascal), energy (joule), power (watt).

• Prefixes: kilo = 10^3, hecto = 10^2, deca = 10, deci = 10^-1, centi = 10^-2, milli = 10^-3, micro = 10^-6, nano = 10^-9.
• Length: 1 km = 1000 m; 1 m = 100 cm; 1 cm = 10 mm. Larger units: 1 light year is about 9.46 x 10^15 m; 1 astronomical unit (AU) is about 1.5 x 10^11 m (Sun to Earth); 1 parsec is about 3.26 light years.
• Time: 1 hour = 3600 s. A simple pendulum or a quartz clock keeps time; an atomic clock is the most accurate.
• Measuring length: a metre scale (least count 1 mm), vernier callipers (0.1 mm or less) and screw gauge (0.01 mm).
• Measuring a curved line: use a thread, then measure the thread against the scale.
• Mass is the amount of matter (kg). Weight is the force of gravity on it (newton).

Motion

An object is in motion if its position changes with time with respect to a reference point. Rest and motion are relative: a passenger is at rest with respect to the bus but moving with respect to the road.

Distance, displacement, speed, velocity

Speed = distance / time Velocity = displacement / time Acceleration a = (v - u) / t

• Scalar quantities have only magnitude; vector quantities have magnitude and direction.
• A body in a full circle has zero displacement but non-zero distance.
• Uniform motion: equal distances in equal times. Non-uniform: unequal distances in equal times.
• Average speed = total distance / total time.
• Retardation (deceleration) is a negative acceleration.
• Conversion: to change km/h to m/s, multiply by 5/18; to change m/s to km/h, multiply by 18/5. So 72 km/h = 20 m/s.
• Instruments: speedometer shows speed; odometer shows distance travelled.

Equations of motion (uniform acceleration)

Here u = initial velocity, v = final velocity, a = acceleration, t = time, s = displacement.

Worked example. A car starts from rest and reaches 20 m/s in 5 s. Then a = (20 - 0) / 5 = 4 m/s^2, and s = 0 + (1/2) x 4 x 25 = 50 m.

Free fall: an object falling only under gravity has a = g = 9.8 m/s^2 (about 10 m/s^2 for quick work). Thrown upward, it slows by g until velocity is zero at the top. A stone dropped from rest falls s = (1/2) g t^2.

Graphs of motion

A horizontal line on a distance-time graph means the body is at rest. A steeper line means higher speed.

Uniform circular motion

A body moving in a circle at constant speed still accelerates, because its direction keeps changing. The force that keeps it in the circle is the centripetal force, directed towards the centre. The tangent to the circle shows the direction of velocity at any point.