Projectile motion

Horizontal motion

$$\begin{align*} v_x &= v_{0x} = v_0 \cos\theta \\ x &= (v_0 \cos\theta) t \end{align*}$$

Vertical motion

$$\begin{align*} v_y &= v_0 \sin\theta - gt \\ y &= (v_0 \sin\theta)t - \frac{1}{2}gt^2 \end{align*}$$

Parabolic trajectory of the projectile

$$y = x\tan\theta - x^2\frac{g}{2v_0^2\cos^2\theta}$$

Time to reach max height:

$$T = \frac{v_0\sin\theta}{g}$$

Total air time:

$$t = 2T$$

Maximum height:

$$H = \frac{v_0^2\sin^2\theta}{2g}$$

Horizontal distance:

$$D = v_xt = \frac{2v_0^2\sin\theta\cos\theta}{g}$$

Center of mass

$$\begin{align*} \vec r_{cm} &= \frac{\sum\limits_i m_i \vec r_i}{M} \\ \vec r_{cm} &= x_{cm} \hat i + y_{cm} \hat j + z_{cm} \hat k \\ \end{align*}$$

• $r_{cm}$: position vector of center of mass
• $M$: total mass of all point masses

Center of gravity

If $\vec g$ is the same at all points on a body, CM = CG and $\vec a_{cm} = \vec g$