2-Dimensional Kinematics Problem: Shooting a dropping coconut

The following is a question (of a more challenging nature) posed to JC1 students when they are studying the topic of kinematics.

A gun is aimed in such a way that the initial direction of the velocity of its bullet lies along a straight line that points toward a coconut on a tree. When the gun is fired, a monkey in the tree drops the coconut simultaneously. Neglecting air resistance, will the bullet hit the coconut?

coconut kinematics
Two-Dimensional Kinematics: Gun and Coconut Problem

It is probably safe to say that if the bullet hits the coconut, the sum of the downward displacement of coconut s_{yc} and the upward displacement of the bullet s_{yb} must be equal to the initial vertical separation between them, i.e. s_{yc}+s_{yb}=H. This is what we need to prove.

Since s_{yc}=\dfrac{1}{2}gt^2s_{yb}=u\text{sin}\theta.t-\dfrac{1}{2}gt^2 and s_{xb}=u\text{cos}\theta t,

s_{yc}+s_{yb}=u\text{sin}\theta\times t=u\text{sin}\theta\times \dfrac{s_{xb}}{u\text{cos}\theta}=s_{xb}\times{\text{tan}\theta}.

At the same time, the relationship between H and the horizontal displacement of the bullet s_{xb} before it reaches the same horizontal position of the coconut is \text{tan}\theta=\dfrac{H}{s_{xb}}.

Hence, s_{yc}+s_{yb}=H!

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