Physics

Physics

Problem

Two stones A and B are projected simultaneously from the same point on level ground with the same initial speed v0v_0. Stone A is thrown at an angle θ\theta with the horizontal while stone B is thrown at an angle (90∘−θ)(90^\circ-\theta) with the horizontal. Neglect air resistance. a. (i) Prove that the horizontal ranges of the two stones are equal. b. (ii) Derive an expression for the ratio of their maximum heights HA/HBH_A/H_B in terms of θ\theta. c. (iii) If v0=20 m s−1v_0 = 20\,\text{m s}^{-1} and θ=30∘\theta = 30^\circ, calculate (a) the common horizontal range, (b) the time of flight of each stone, and (c) the individual maximum heights. Take g=9.8 m s−2g = 9.8\,\text{m s}^{-2}.
Intuitive explanation

The key insight is recognizing that these two launch angles are complementary—they're reflections of each other across 45°. Even though the stones follow different paths, projectile motion's symmetry means they share the same range and time of flight. Focus on how the initial velocity components trade places: what A gains in horizontal speed, B gains in vertical speed. This lets you find relationships between their trajectories without solving the full equations.

Intuitive explanation

The key insight is recognizing that these two launch angles are complementary—they're reflections of each other across 45°. Even though the stones follow different paths, projectile motion's symmetry means they share the same range and time of flight. Focus on how the initial velocity components trade places: what A gains in horizontal speed, B gains in vertical speed. This lets you find relationships between their trajectories without solving the full equations.

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