The correct answer is B. R² = 16 h₁h₂.
Why is R² = 16 h₁h₂ correct?
Both particles have the same initial speed (u) and the same range (r). This means they both reach the same horizontal distance before falling back down.
However, they have different maximum heights (h₁ and h₂). This implies they launched at different angles: particle with h₁ has a steeper angle and the one with h₂ has a shallower angle
Range (r) = u *
(2) * sin(θ) * cos(θ)
Maximum height (h) = u * sin²(θ) / 2g
Combining these equations for each particle and taking the ratio of their maximum heights (h₁/h₂):
h₁/h₂ = tan²(θ₁) / tan²(θ₂) = (90-θ₁)² / (90-θ₂)²
Since θ₁ and θ₂ are close to 45°, their complements (90-θ₁ and 90-θ₂) will be close to 45° as well. Therefore, we can approximate the expression as:
h₁/h₂ ≈ (45 + x)² / (45-x)² ≈ (1 + x/45)² ≈ 1 + 2x/45 + x²/2025
where x is a small angle deviation from 45°.
(h₁/h₂)² ≈ 1 + 4x/45
Now, substitute r from the first equation back into the expression for h₁/h₂:
h₁/h₂ = (u *
2) * sin(θ) * cos(θ))/ (u² * sin²(θ) / 2g) = 2
2) / sin(θ)
sin²(θ) ≈ 1 / (4 + 8x/45) ≈ 45 / (45 + 8x)
sin²(θ) ≈ 1 - 8x/45
r² = u² * 2 * (1 - 8x/45) = (2u²) * (22.5 - 4x/45) ≈ 45u² - 4u² * x/22.5
Therefore, the answer is B. R² = 16 h₁h₂.