Projectile just reaches the top of a vertical column A jet is projected at 45° from a point 40 m away from the foot of a vertical column and just reaches the top of the column. Assuming ideal projectile motion, what is the height of the column?

Introduction / Context:
Projectile kinematics is widely applied to free jets from nozzles. When a projectile “just reaches” an obstacle, the apex (maximum height) of the parabolic trajectory commonly coincides with the obstacle’s top for the critical case.

Given Data / Assumptions:

• Launch angle θ = 45°.
• Horizontal distance from launch point to column foot = 40 m.
• Neglect air resistance.
• Critical case where the jet apex grazes the top of the column.

Concept / Approach:
For a projectile, the horizontal coordinate of the apex is x_apex = (u^2 * sin 2θ) / (2 g). For θ = 45°, sin 2θ = 1, so x_apex = u^2 / (2 g). The maximum height is H_max = (u^2 * sin^2 θ) / (2 g) = u^2 / (4 g) for θ = 45°. If the column is exactly at the apex, x_apex = 40 m and its height equals H_max.

Step-by-Step Solution:

Verification / Alternative check:
Using the trajectory equation y = x tan θ − (g x^2) / (2 u^2 cos^2 θ) at x = 40 m with θ = 45° yields the same height.

Why Other Options Are Wrong:

• 15 m and 30 m: Do not satisfy apex relations for θ = 45° at x = 40 m.
• 40 m and 60 m: Far exceed the maximum height given the apex location constraint.

Common Pitfalls:
Assuming the obstacle is not at the apex; forgetting sin 2θ = 1 at 45°; mixing metres with other units.

Final Answer:
20 m