The angle of projection of a particle is measured from the vertical axis as φ and the maximum height reached by the particle is hₘ. Here hₘ as function of φ can be presented as

Solution

Physics sir · 4 steps

The angle is from the vertical, so the vertical velocity is u cos φ; check the two ends of the graph.

1. Vertical component. With φ measured from the vertical, the upward component of velocity is u cos φ. Only this component decides how high the particle rises.

   hₘ = (u²cos² φ)/2g

2. Check the ends. At φ = 0° the particle goes straight up, so hₘ = u²/2g, the largest possible value. At φ = 90° it moves horizontally, so hₘ = 0.
3. Shape in between. cos² φ falls steadily as φ goes from 0° to 90°. Its slope is -sin 2φ, which is zero at φ = 0°, so the curve leaves the hₘ axis flat and then bends down to meet the φ axis.
4. So. The graph must start at a finite maximum on the hₘ axis and come down to zero at 90°. Only graph (d) does this.

The idea to keep. For any graph question, first check where the curve starts and where it ends. Then watch the angle convention: from the vertical, the upward component is u cos φ, not u sin φ.

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