The speed of a particle with velocity v(t) = (dx/dt, dy/dt) is which expression?

• A. (dx/dt) + (dy/dt)
• B. sqrt((dx/dt)^2 + (dy/dt)^2)
• C. sqrt((d^2x/dt^2)^2 + (d^2y/dt^2)^2)
• D. sqrt((dx/dt)^2 - (dy/dt)^2)

Answer: (B)

Speed is the length (magnitude) of the velocity vector. For a particle moving in the plane with velocity v(t) = (dx/dt, dy/dt), the speed is the Euclidean norm of that vector: |v(t)| = sqrt((dx/dt)^2 + (dy/dt)^2). This comes from the idea that the velocity components form a vector, and its length in the plane is given by the square root of the sum of the squares of the components (the dot product v·v = (dx/dt)^2 + (dy/dt)^2).

The other forms don’t measure the length of the velocity vector: adding the components gives a scalar that mixes the two directions; using second derivatives would relate to acceleration, not velocity; and taking a difference inside the square root doesn’t represent the true length in Euclidean space. In three dimensions, the same idea extends to sqrt((dx/dt)^2 + (dy/dt)^2 + (dz/dt)^2).