Two bodies are projected with the same velocity. If one is projected at an angle of 30° and the other at an angle of 60° to the horizontal, the ratio of the maximum heights reached is

The correct option is 1 : 3.

Understanding the Concept:
When a body is projected with an initial velocity u at an angle θ with the horizontal, it follows a parabolic trajectory. The maximum height (H) reached by the projectile is given by the formula:

$H=\frac{u^2{\sin }^2\theta }{2g}$

where:
- u is the initial velocity of projection,
- θ is the angle of projection with the horizontal, and
- g is the acceleration due to gravity.

Given that both bodies are projected with the same velocity (u) and at the same location (so g is constant), the maximum height is directly proportional to the square of the sine of the angle of projection:

$H\propto {\sin }^2\theta$

Therefore, the ratio of their maximum heights is:

$\frac{H_1}{H_2}=\frac{{\sin }^2{\theta }_1}{{\sin }^2{\theta }_2}$

Step-by-Step Calculation:
The given angles of projection are:

${\theta }_1=30°$

and

${\theta }_2=60°$

Substituting the values of the angles into the ratio formula:

$\frac{H_1}{H_2}=\frac{{\sin }^2(30°)}{{\sin }^2(60°)}$

We know the values of sine for these standard angles:

$\sin (30°)=\frac{1}{2}$

and

$\sin (60°)=\frac{\sqrt{3}}{2}$

Now, calculate the squares of these values:

${\sin }^2(30°)={\left(\frac{1}{2}\right)}^2=\frac{1}{4}$

and

${\sin }^2(60°)={\left(\frac{\sqrt{3}}{2}\right)}^2=\frac{3}{4}$

Substitute these squared values back into the ratio equation:

$\frac{H_1}{H_2}=\frac{1/4}{3/4}=\frac{1}{3}$

Thus, the ratio of the maximum heights reached by the two bodies is 1 : 3.