Section 3

2D and 3D Motion

r = x \hat{i} + y \hat{j} + z \hat{k}

Δ r = r_{f} - r_{i}

v_{med} = \frac{Δ r}{Δ t}

v = \frac{d r}{d t}

a = \frac{d v}{d t} = \frac{d ^{2} r}{d t ^{2}}

v_{0 x} = v_{0} cos θ, v_{0 y} = v_{0} sin θ

x = x_{0} + (v_{0} cos θ) t

y = y_{0} + (v_{0} sin θ) t - \frac{1}{2} g t^{2}

v_{y} = v_{0} sin θ - g t

Note: this corrects the expression $v_{y} = v_{0} sin θ$ , which corresponds only to the initial value.

v_{x} = v_{0} cos θ

t_{m a x} = \frac{v _{0} sin θ}{g}

H = \frac{v _{0}^{2} sin ^{2} θ}{2 g}

additional height with respect to the launch point.

t_{vuelo} = \frac{2 v _{0} sin θ}{g}

R = \frac{v _{0}^{2} sin ( 2 θ )}{g}

same initial and final height, horizontal surface, and negligible air resistance.

v_{P / A} = v_{P / B} + v_{B / A}