Vectors and motion in two dimensions

Two-dimensional vector problems reduce to pairs of one-dimensional problems once vectors are resolved into perpendicular components. The cosine function projects a vector onto the axis from which the angle is measured, and the sine function projects it onto the perpendicular axis.

Component Method

1. Resolve each vector into its x- and y-components
2. Sum the x-components and y-components separately
3. Reconstruct the resultant's magnitude using the Pythagorean theorem and its direction using the inverse tangent

This component method is exact and works for any number of vectors at any angles.

Projectile Motion

Projectile motion applies this same independence of axes to kinematics. With zero horizontal acceleration and constant vertical acceleration g directed downward, the horizontal and vertical motions are solved independently and linked through the shared time of flight.

• The horizontal velocity remains constant
• The vertical velocity changes by g each second
• At the peak, v_y = 0 but the acceleration is still g downward

Recognizing these features — constant vₓ, changing v_y, shared t, and the parabolic trajectory that results — is essential for every projectile problem on the AP exam.