<p align="right">Last Update: <font color="#4f81bd">December 11, 2024</font></p> Determine the range without using values. --- ![[86E00BA8-1EAA-4DDB-9BE1-29EFECA14F75.jpeg]] ### Horizontal $\Delta x$ = range $\vec{v}_x$ = horizontal velocity $t$ = flight time Formula for range $\Delta{\vec {x}} = \vec{v}{_x} \cdot t$ --- Find $t_{\frac{1}{2}}$ (time to peek) $\vec{v}{_{1y}} = \vec{v}_{0y} + \vec{a}t_\frac{1}{2}$ --- Rearrange for $t_{\frac{1}{2}}$ $t_{\frac{1}{2}} = \frac{\vec{v}_{1y}-\vec{v}_{0y}}{a}$ --- Substitute in values $t_{\frac{1}{2}} = \frac{0-\vec{v}\cdot sin\theta}{-g}$ $t_{\frac{1}{2}} = \frac{\vec{v}\cdot sin\theta}{g}$ --- Double time to get to time of flight $t = \frac{2\cdot\vec{v}\cdot sin\theta}{g}$ --- Substitute time of flight found above into formula for range $\Delta{\vec {x}} = \vec{v_x} \cdot \frac{2\cdot\vec{v}\cdot sin\theta}{g}$ --- Substitute $\vec{v_x}$ with $\vec{v}\cdot\cos\theta$ $\Delta{\vec {x}} = \vec{v}\cdot\cos\theta \cdot \frac{2\cdot\vec{v}\cdot sin\theta}{g}$ --- Algebraically combine values $\Delta{\vec {x}} = \frac{\vec{v} {^2}\cdot sin 2\theta}{g}$ ## Video <iframe width="100%" height="480px" style="border:0;" scrolling="no" src="https://go.screenpal.com/player/cZh1D6VMKH4?width=100%&height=100%&ff=1&title=0" allowfullscreen="true"></iframe> ### Related Topics --- Return [[Home|Home]] | [[Notes Vault/Physics Notes Vault/Kinematics/2D Motion/Projectiles|Projectiles]] | [[Time of Flight]] | [[Find Max Height|Maximum Height]] | [[Find Range|Range]]