Inclined jet kinematics – height vs horizontal span A water jet is issued at speed u making an angle θ with the horizontal. The maximum height H it reaches and the total horizontal span (range) R are measured. What is the ratio H / R expressed in terms of θ (neglect air resistance)?

Introduction / Context:
Projectile motion under constant gravity provides useful analogies for free jets from nozzles. Relating the jet’s peak height to its horizontal reach helps in setting spray distances and shielding for hydraulic jets and fountains.

Given Data / Assumptions:

• Initial speed u at launch angle θ above horizontal.
• Uniform gravitational acceleration g; air drag neglected.
• Ground at same elevation as launch point.

Concept / Approach:

Standard projectile relations: maximum height H = u^2 sin^2θ / (2 g) and range R = u^2 sin 2θ / g. Taking the ratio removes dependence on u and g, leaving a pure function of θ.

Step-by-Step Solution:

Verification / Alternative check:

At θ = 45°, H/R = (1/4) tan 45° = 1/4, which is consistent since the 45° shot maximizes R and yields a moderate H relative to R.

Why Other Options Are Wrong:

sin θ, cos θ, tan θ, and cot θ ignore the factor 1/4 that arises from combining the height and range formulas.

Common Pitfalls:

Using degrees of trigonometric identities incorrectly; forgetting that sin 2θ = 2 sinθ cosθ, which introduces the factor 4 in the denominator.

Final Answer:

(1/4) tan θ