Two projectiles thrown at 30° and 45° with the horizontal respectively, reach the maximum height in same time. The ratio of their initial velocities is :

Solution

Physics sir · 3 steps

Same time to the top means the same vertical launch speed, so set u sin θ equal for both.

1. Time to the top. At maximum height the vertical velocity becomes zero. Gravity slows the vertical component u sin θ at the rate g, so the time to reach the top is t = (u sin θ)/g.
2. Equal times. Both projectiles take the same time and g is the same for both. So their vertical launch components must be equal.

   u₁ sin 30° = u₂ sin 45° ⇒ u₁/u₂ = (sin 45°)/(sin 30°) = (1/√2)/(1/2) = √2

3. So. The projectile at 30° must be thrown faster, and u₁ : u₂ = √2 : 1, which is option (c).

The idea to keep. The time to reach the top depends only on the vertical component u sin θ. Equal times means equal u sin θ, so the lower angle needs the larger speed.

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