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Speed

Speed

Distance covered per unit of time. Speed is a nonnegative scalar. For motion in one dimension, such as on a number line, speed is the absolute value of velocity. For motion in two or three dimensions, speed is the magnitude of the velocity vector.

Key Formula

speed=dt\text{speed} = \frac{d}{t}
Where:
  • dd = Distance traveled (always nonnegative)
  • tt = Elapsed time (must be positive)

Worked Example

Problem: A car travels 150 km in 2 hours. What is its average speed?
Step 1: Identify the distance and time.
d=150 km,t=2 hd = 150 \text{ km}, \quad t = 2 \text{ h}
Step 2: Apply the speed formula.
speed=dt=1502=75 km/h\text{speed} = \frac{d}{t} = \frac{150}{2} = 75 \text{ km/h}
Answer: The car's average speed is 75 km/h.

Another Example

Problem: An object moves in two dimensions with a velocity vector of v⃗=⟨3,−4⟩\vec{v} = \langle 3, -4 \rangle m/s. Find its speed.
Step 1: Recall that speed is the magnitude of the velocity vector.
speed=∣v⃗∣=vx2+vy2\text{speed} = |\vec{v}| = \sqrt{v_x^2 + v_y^2}
Step 2: Substitute the components.
speed=32+(−4)2=9+16=25=5 m/s\text{speed} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \text{ m/s}
Answer: The object's speed is 5 m/s. Notice that although one velocity component is negative, the speed itself is positive.

Frequently Asked Questions

What is the difference between speed and velocity?
Speed is a scalar — it only tells you how fast something moves. Velocity is a vector — it tells you both how fast and in what direction. For example, 60 km/h is a speed, while 60 km/h north is a velocity. Speed is always the absolute value (1D) or magnitude (2D/3D) of velocity, so it can never be negative.
Can speed ever be negative?
No. Speed is defined as a nonnegative quantity. Even if an object moves backward (negative velocity in one dimension), its speed is the absolute value of that velocity, which is positive or zero.

Speed vs. Velocity

Speed is a nonnegative scalar that measures only how fast an object moves. Velocity is a vector that includes both magnitude and direction. In one dimension, speed equals the absolute value of velocity: speed=∣v∣\text{speed} = |v|. In two or three dimensions, speed equals the magnitude of the velocity vector: speed=∣v⃗∣\text{speed} = |\vec{v}|. An object traveling at 5 m/s to the left has a velocity of −5-5 m/s (in a coordinate system where right is positive), but its speed is simply 5 m/s.

Why It Matters

Speed appears throughout mathematics and science whenever you model motion — from basic rate problems in algebra to calculus-based kinematics. Understanding that speed strips away directional information is essential when setting up distance–rate–time equations, computing arc lengths, or analyzing the magnitude of a derivative. In everyday life, speedometers, speed limits, and athletic records all rely on this concept.

Common Mistakes

Mistake: Treating speed and velocity as interchangeable.
Correction: Speed is a scalar (no direction, never negative), while velocity is a vector (has direction, can be negative in one dimension). Always check whether a problem asks for speed or velocity.
Mistake: Thinking speed can be negative.
Correction: Speed is defined as the absolute value or magnitude of velocity, so it is always zero or positive. A negative sign indicates direction, which belongs to velocity, not speed.

Related Terms

  • Velocity — Vector quantity whose magnitude equals speed
  • Absolute Value — Gives speed from 1D velocity
  • Magnitude — Gives speed from a velocity vector
  • Scalar — Speed is a scalar, not a vector
  • Vector — Velocity is a vector; speed is its magnitude
  • Nonnegative — Speed is always nonnegative
  • Number Line — Used for one-dimensional motion