# Curvilinear Translation (Projectile Motion)

Projectile motion follows a parabolic trajectory. The vertical component of projectile is under constant gravitational acceleration and the horizontal component is at constant velocity. For easy handling, resolve the motion into x and y components and use the formulas in rectilinear translation.

Form the figure below:

$v_{oy} = v_o \, \sin \theta$

At any point B

_{x}= 0.

$x = v_{ox}t$

For the y-component of motion, a_{y} = -g. Notice that the first three formulas that follow are taken from motion with constant acceleration.

$y = v_{oy}t - \frac{1}{2}gt^2$

${v_{By}}^2 = {v_{oy}}^2 - 2gy$

From x = v_{ox}t, t = x/v_{ox}. Substitute t = x/v_{ox} to y = v_{oy}t - ½ gt^{2}.

$y = v_{oy}\left( \dfrac{x}{v_{ox}} \right) - \frac{1}{2}g\left( \dfrac{x}{v_{ox}} \right)^2$

$y = v_{oy}\left( \dfrac{x}{v_{ox}} \right) - \frac{1}{2}g\left( \dfrac{x^2}{{v_{ox}}^2} \right)$

$y = v_o \, \sin \theta \left( \dfrac{x}{v_o \, \cos \theta} \right) - \frac{1}{2}g\left[ \dfrac{x^2}{(v_o \, \cos \theta)^2} \right]$

$y = \dfrac{x \, \sin \theta}{\cos \theta} - \dfrac{gx^2}{2{v_o}^2 \, \cos^2 \theta}$

At point A

_{Ay}= 0.

$t = \dfrac{v_{oy}}{g}$

At point C

_{C}= v

_{o}, and v

_{y}= -v

_{oy}

$t = \dfrac{2v_{oy}}{g} = \dfrac{2v_o \, \sin \theta}{g}$

Note:

- v
_{y}is positive if directed upward and negative if directed downward - At any point D below the origin O, the sign of y is negative.