Velocity and Speed

• The precise difference between speed (scalar) and velocity (vector)
• How displacement differs from distance
• Average speed vs instantaneous speed — and when each matters
• Why a GPS uses velocity, not just speed

Distance and Displacement Distance is the total length of the path travelled by an object, regardless of direction. It is a scalar quantity (magnitude only), always positive or zero.

Displacement is the shortest straight-line distance from the starting point to the ending point, along with the direction. It is a vector quantity. Displacement can be zero even if the distance is large — if you return to your starting point, your displacement is zero.

Example: An athlete runs one complete lap (400 m) of a circular track. Distance = 400 m. Displacement = 0 m (they start and finish at the same point).

Speed Speed = distance / time. It tells you how fast an object is moving but nothing about direction. It is a scalar quantity measured in m/s or km/h.

Speed = d / t

Average speed = total distance / total time (useful for journeys with varying speed). Instantaneous speed = speed at a particular instant (what your car's speedometer shows).

Velocity Velocity = displacement / time. It tells you how fast and in which direction. It is a vector quantity.

Velocity = displacement / time

Average velocity = total displacement / total time. Instantaneous velocity = velocity at a specific instant (magnitude equals instantaneous speed, plus direction).

An object can have a non-zero speed but zero average velocity (the 400 m runner above). An object can also change velocity without changing speed — a car moving in a circle at constant speed has changing velocity because direction is continuously changing.

Relative velocity The velocity of object A relative to object B is: v_AB = v_A - v_B

Two trains moving in the same direction at 60 km/h and 40 km/h: the faster train's velocity relative to the slower = 60 - 40 = 20 km/h. If moving in opposite directions: 60 + 40 = 100 km/h (they approach each other very quickly).

Why GPS uses velocity A GPS receiver calculates your position by timing signals from satellites. Navigation requires knowing which way you are going (direction), not just how fast. If you are 500 m from a junction, the GPS needs to know your direction of travel to tell you when to turn. Speed alone (a scalar) cannot provide turn-by-turn navigation — velocity (vector) can.

Units and conversion 1 m/s = 3.6 km/h To convert km/h to m/s: divide by 3.6 To convert m/s to km/h: multiply by 3.6

Problem: A person walks 3 km East, then 4 km North. The journey takes 2 hours. Find: (a) Total distance covered. (b) Magnitude of displacement. (c) Average speed. (d) Magnitude of average velocity.

Solution:

(a) Total distance = 3 + 4 = 7 km

(b) Displacement uses the shortest straight-line path. The two legs form a right angle (East then North), so: |displacement| = √(3² + 4²) = √(9 + 16) = √25 = 5 km Direction: North-East (at an angle — specifically arctan(4/3) ≈ 53° from East, toward North)

(c) Average speed = total distance / time = 7 km / 2 h = 3.5 km/h

(d) Average velocity = displacement / time = 5 km / 2 h = 2.5 km/h (directed North-East)

Notice: average speed (3.5 km/h) ≠ magnitude of average velocity (2.5 km/h) in this case.

• Confusing distance with displacement. Displacement is the vector from start to end; distance is total path length. They are equal only for straight-line one-way travel.
• Saying average velocity = (u + v)/2 always. This formula is only valid for uniform acceleration. In general, average velocity = total displacement / total time.
• Forgetting direction for velocity answers. If the question asks for velocity, the answer must include magnitude and direction. Just "5 m/s" is speed, not velocity.
• Speed and velocity have different averages. Average speed uses total distance; average velocity uses total displacement. They are not the same quantity.

1. A car travels 60 km East and then 60 km West in 3 hours total. What is the average speed and average velocity?
2. Convert 90 km/h to m/s.
3. A ball moves in a circle of radius 7 m, completing one full circle in 4 s. What is the average velocity over one complete revolution? What is the average speed?