Velocity and Speed

Speed and velocity sound alike but are not interchangeable. Speed is a scalar; velocity is a vector. The distinction lies at the heart of kinematics.

Concept

Speed

• Average speed = total distance / total time.
• Instantaneous speed = magnitude of instantaneous velocity.
• Always $\ge 0$.

Velocity

• Average velocity = displacement / time interval: $\bar{v} = \frac{\Delta x}{\Delta t}.$
• Instantaneous velocity = derivative of position: $v(t) = \lim_{\Delta t \to 0}\frac{\Delta x}{\Delta t} = \frac{dx}{dt}.$
• A vector; can be negative in 1D (indicates direction).

Key Relations

For motion entirely in one direction, average speed = $|\bar{v}|$. Otherwise speed $\ge |\bar{v}|$.

Worked Example

Q: A car travels from A to B (60 km) at 60 km/h, then from B to A at 40 km/h. Find the average speed and average velocity for the round trip.

Solution: Time from A to B: $t_1 = 60/60 = 1$ h. Time from B to A: $t_2 = 60/40 = 1.5$ h. Total distance: $60 + 60 = 120$ km. Total displacement: $0$ (returned to A). Total time: $2.5$ h.

• Average speed = $120/2.5 = 48$ km/h.
• Average velocity = $0/2.5 = 0$.

This is also the harmonic mean formula: $\bar{v} = \frac{2 v_1 v_2}{v_1 + v_2} = \frac{2 \cdot 60 \cdot 40}{60+40} = \frac{4800}{100} = 48\,\text{km/h}.$

(Note: this formula applies for equal distances in both legs.)

Common Confusions

• "Speed is the magnitude of velocity." — True for instantaneous speed/velocity. The average speed is NOT in general the magnitude of average velocity.
• "Negative velocity means slowing down." — No. It means moving in the negative direction. Slowing down corresponds to deceleration (opposite-sign velocity and acceleration).
• "If average velocity is zero, the object was stationary." — Wrong; a round trip gives $\bar{v} = 0$ but motion happened.

Key Takeaways

• Speed is scalar (never negative); velocity is a vector.
• Average velocity uses displacement; average speed uses distance.
• Instantaneous velocity is the time derivative of position.
• Sign of velocity in 1D indicates direction, not magnitude of slowdown.