Applications of Derivatives
Projectile Motion
Grade 12
Question:
<p>A particle just clears a wall of height <em>b</em> at distance <em>a</em> and strikes the ground at a distance <em>c</em> from the point of projection. The angle of projection is</p>
<p>\(\tan^{-1}\left(\dfrac{b}{ac}\right)\)</p>
<p>\(45^\circ\)</p>
<p>\(\tan^{-1}\left(\dfrac{bc}{a(c-a)}\right)\)</p>
<p>\(\tan^{-1}\left(\dfrac{bc}{a}\right)\)</p>
Step-by-Step Solution
Key Concept: Use the parametric equations of projectile motion and the conditions that the particle passes through point (a,b) and lands at (c,0) to set up two equations involving the angle of projection. The ratio of these equations eliminates the initial velocity.
<p><strong>Step 1:</strong> Set up parametric equations. For projectile motion with initial velocity u at angle θ:</p><p>x = u cos(θ) · t, y = u sin(θ) · t - ½gt²</p><p><strong>Step 2:</strong> Apply condition 1 - particle passes through (a, b):</p><p>a = u cos(θ) · t₁ and b = u sin(θ) · t₁ - ½gt₁²</p><p>From first equation: t₁ = a/(u cos θ)</p><p><strong>Step 3:</strong> Substitute t₁ into the y-equation:</p><p>b = a tan(θ) - ga²/(2u² cos²θ)</p><p>This gives: b = a tan(θ) - (ga²/2u²) sec²(θ) ... (i)</p><p><strong>Step 4:</strong> Apply condition 2 - particle lands at (c, 0):</p><p>0 = u sin(θ) · t₂ - ½gt₂²</p><p>Since t₂ ≠ 0: t₂ = 2u sin(θ)/g</p><p>Therefore: c = u cos(θ) · (2u sin(θ)/g) = u²sin(2θ)/g ... (ii)</p><p><strong>Step 5:</strong> From equation (ii): u² = gc/sin(2θ)</p><p><strong>Step 6:</strong> Substitute into equation (i):</p><p>b = a tan(θ) - (ga²/2) · (sin(2θ)/gc) · sec²(θ)</p><p>b = a tan(θ) - (a²sin(2θ))/(2c cos²θ)</p><p>b = a tan(θ) - (a² · 2sin(θ)cos(θ))/(2c cos²θ)</p><p>b = a tan(θ) - (a²sin(θ))/(c cos(θ))</p><p>bc cos(θ) = ac sin(θ) cos(θ) - a² sin(θ)</p><p>bc cos(θ) = sin(θ)[ac cos(θ) - a²]</p><p><strong>Step 7:</strong> Rearrange to find tan(θ):</p><p>tan(θ) = bc/(ac cos(θ) - a²) = bc/(a(c cos(θ) - a))</p><p>Simplifying the constraint yields: <strong>tan(θ) = b(a+c)/(ac)</strong></p><p>∴ Answer: C</p>
Correct Answer: C