Two projectiles are projected at 30° and 60° with the horizontal with the same speed. The ratio of the maximum height attained by the two projectiles respectively is:

Solution

Physics sir · 4 steps

Same speed, so H depends only on sin squared of the angle: compare sin² 30° with sin² 60°.

1. Height formula. Only the vertical part of the velocity, u sin θ, decides how high the projectile rises. At the top it has fallen to zero, which gives the formula below.

   H = (u² sin² θ)/2g

2. Take the ratio. Both projectiles have the same u and the same g, so u²/2g cancels. That leaves H₁/H₂ = (sin² 30°)/(sin² 60°).
3. Put in values. sin² 30° = (1/2)² = 1/4 and sin² 60° = (√3/2)² = 3/4. Their ratio is (1/4)/(3/4) = 1/3.
4. So. The maximum heights are in the ratio 1 : 3, option (c).

The idea to keep. For the same launch speed, H ∝ sin² θ. Do not forget the square: it is the step that turns a ratio of sines into a ratio of heights.

Answer from the NTA final answer key. Solution by PrepWiser.