Scalars and Vectors

In physics, quantities are classified as either scalar or vector.

• A scalar has magnitude only. Examples: speed (50 km/h), temperature (25 °C), mass (70 kg), time (10 s).
• A vector has both magnitude and direction. Examples: velocity (50 km/h north), displacement (3 m east), force (10 N downward).

Distance vs. Displacement

Distance is the total length of the path travelled — it is a scalar. Displacement is the straight-line distance from start to finish, including direction — it is a vector.

Example: A runner completes one lap of a 400 m track. Distance = 400 m. Displacement = 0 m, because they are back at the starting point.

Speed and Velocity

Speed is how fast an object moves — the rate of change of distance. It is a scalar. Velocity is the rate of change of displacement, including direction — it is a vector.

$$v = \frac{\Delta x}{\Delta t}$$

where $\Delta x$ is displacement (m) and $\Delta t$ is the time interval (s). The SI unit of velocity is metres per second ($\text{m/s}$).

Example: A car travels 150 m east in 10 s. Its velocity is $v = 150 / 10 = 15\ \text{m/s east}$. Its speed is $15\ \text{m/s}$ (magnitude only, no direction stated).

Acceleration

Acceleration is the rate of change of velocity:

$$a = \frac{\Delta v}{\Delta t}$$

where $\Delta v$ is the change in velocity and $\Delta t$ is the elapsed time. The SI unit is metres per second squared ($\text{m/s}^2$). Acceleration is a vector — it has direction. An object can accelerate by changing speed, changing direction, or both.

Example: A runner starts from rest ($0\ \text{m/s}$) and reaches $8\ \text{m/s}$ in 4 s. Acceleration $= (8 - 0) / 4 = 2\ \text{m/s}^2$ in the direction of motion.