The instantaneous speed of a particle moving with position r(t) = (x(t), y(t)) is given by which expression?

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

Answer: (A)

Speed is the magnitude of the velocity. For a particle with position r(t) = (x(t), y(t)), the velocity is the derivative r′(t) = (dx/dt, dy/dt). The speed is the length of this vector, which is sqrt((dx/dt)^2 + (dy/dt)^2) by the Pythagorean theorem.

The other forms don’t match:.sqrt((dx/dt)^2 − (dy/dt)^2) isn’t a general length and can be invalid; dx/dt + dy/dt is a scalar that isn’t the magnitude of the velocity; and (dx/dt, dy/dt) is the velocity vector itself, not its speed.