2D Kinematics: Velocity Component Decomposition

How do the x and y components of velocity give speed and direction?

In short: In 2D, velocity is a vector with a horizontal part vx and a vertical part vy. Its speed is the length of that vector, |v| = √(vx² + vy²), which is never negative. Its direction is the angle θ measured from the +x axis, found from tan θ = vy/vx. The two components change independently, so you can follow each one on its own and then combine them.

Common doubts

• “If vx is negative, is the speed negative too?” No. A negative vx only tells you the object is moving toward −x. Speed is the magnitude √(vx² + vy²), so it is always zero or positive.
• “What happens to speed when the object turns around?” Speed does not change sign. A component passes through zero as it reverses, and the speed is zero only at an instant when both vx and vy are zero.
• “Does θ = tan⁻¹(vy/vx) always give the right angle?” Not always. A calculator returns angles only between −90° and +90°, so when vx is negative you must add 180° or check the quadrant from the signs of vx and vy.

Use PAUSE and STEP in the simulation above to freeze the motion and watch vx, vy and |v| change together.

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