Motion in a Plane — Formula Sheet

|\vec{R}| = \sqrt{A^2 + B^2 + 2AB\cos\theta}

\tan\alpha = \frac{B\sin\theta}{A + B\cos\theta}

\vec{A} \cdot \vec{B} = AB\cos\theta

|\vec{A} \times \vec{B}| = AB\sin\theta

h_{\max} = \frac{u^2 \sin^2\theta}{2g}

T = \frac{2u \sin\theta}{g}

R = \frac{u^2 \sin 2\theta}{g}

y = x\tan\theta - \frac{gx^2}{2u^2\cos^2\theta}

a_c = \frac{v^2}{r}

F_c = \frac{mv^2}{r} = mr\omega^2

v = r\omega

Horizontal motion

constant velocity (ax = 0).

Magnitude of centripetal acceleration

a = v²/r.

Dot product

A·B = AB cosθ — scalar result

Cross product

A×B = AB sinθ — vector result, perpendicular to A and B

For a vector A at angle θ, Ax = A cosθ and Ay = A sinθ.

By eliminating time from the equations of motion, we get y = x tanθ − (gx²)/(2u²cos²θ), which is a parabolic equation.