Calculus 1 — 22.6: Projectile Motion Using Integration скачать в хорошем качестве

Calculus 1 — 22.6: Projectile Motion Using Integration

Where does the equation s(t) = −16t² + v₀t + s₀ actually come from? Instead of memorizing it, this video derives the classic projectile motion formula from scratch — starting with nothing but gravitational acceleration and two initial conditions. Through two successive integrations, you'll see how each constant of integration carries physical meaning and how initial conditions transform a general antiderivative into a specific solution. Key concepts covered: • The derivative-integral chain: position, velocity, and acceleration connected by differentiation and integration • Integrating constant acceleration (a = −32 ft/s²) to recover velocity, then integrating velocity to recover position • The role of the constant of integration (+C) and why omitting it gives the wrong physical scenario • Using initial conditions v(0) = 128 and s(0) = 16 to pin down specific solutions • Physical meaning of each term: −16t² (gravity), v₀t (initial velocity), s₀ (initial height) • Finding maximum height by setting velocity equal to zero (optimization via the derivative) • Finding time of ground impact by setting position equal to zero and applying the quadratic formula • Interpreting negative time solutions and discarding non-physical results • The general pattern: any vertical projectile problem requires just three inputs (acceleration, initial velocity, initial height) • Practice problem: deriving maximum height for a ball launched from ground level at 64 ft/s ━━━━━━━━━━━━━━━━━━━━━━━━ SOURCE MATERIALS The source materials for this video are from    • Calculus 1 Lecture 4.1:  An Introduction t...