The ranges and heights for two projectiles projected with the same initial velocity at angles 42° and 48° with the horizontal are R₁, R₂ and H₁, H₂ respectively. Choose the correct option:

CONCEPT:

Projectile Motion

• When two projectiles are launched with the same initial velocity but at different angles, their ranges and maximum heights can be compared.
• The range (R) of a projectile is given by the formula: R = (v² × sin(2θ)) / g where:
  • v is the initial velocity
  • θ is the angle of projection
  • g is the acceleration due to gravity
• The maximum height (H) of a projectile is given by the formula: H = (v² × sin²(θ)) / (2g)

EXPLANATION:

• Let's examine the given options:
  • Option 1: R₁ > R₂ and H₁ = H₂
    • This option suggests the range of the first projectile is greater than that of the second, and their maximum heights are equal. This is incorrect because if the angles of projection are different, their maximum heights will also differ.
  • Option 2: R₁ = R₂ and H₁ < H₂
    • This option suggests the ranges of both projectiles are equal, but the height of the first is less than that of the second. This is correct. When two angles are complementary (e.g., 42° and 48°), the projectiles have equal ranges but different maximum heights.
  • Option 3: R₁ < R₂ and H₁ < H₂
    • This is incorrect because complementary angles result in equal ranges, so R₁ cannot be less than R₂ in this context.
  • Option 4: R₁ = R₂ and H₁ = H₂
    • This option assumes both range and height are equal, which is incorrect as maximum height depends on the square of sine of the angle, and hence varies with angle.

Therefore, the correct answer is: Option 2 — R₁ = R₂ and H₁ < H₂.