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Home » Use the formula for the sum of the first n natural numbers: Sum = n(n + 1)/2 - Silent Sales Machine

Use the formula for the sum of the first n natural numbers: Sum = n(n + 1)/2 - Silent Sales Machine

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 A science communicator is creating an educational video about projectile motion. Suppose a projectile is launched from the ground with an initial velocity $v_0$ at an angle $\theta$ to the horizontal. Given the equation of the trajectory $y = x\tan\theta - \frac{g}{2v_0^2\cos^2\theta}x^2$, find the horizontal distance $x$ at which the projectile reaches its maximum height.

📌 The trajectory equation is given by

📌 \[ y = x\tan\theta - \frac{g}{2v_0^2\cos^2\theta}x^2. \]

📌 This is a quadratic equation in terms of $x$. For a quadratic equation of the form \( y = ax^2 + bx + c \), the $x$-value at which the maximum (or minimum) occurs is given by

📌 \[ x = -\frac{b}{2a}. \]

📌 \[ a = -\frac{g}{2v_0^2\cos^2\theta}, \quad b = \tan\theta. \]

📌 Thus, the horizontal distance $x$ at which the projectile reaches its maximum height is

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