Animated Solution for Mathematics - Differentiation: A body falling from rest under gravity passes a certain point P. It was at a distance of 400 m from P, 4s prior to passing through P. If $g=10{\text{ m/s}}^2$, then the height above the point P from where the body began to fall is

Imagine you are standing on the edge of a precipice, watching a body fall freely under the influence of gravity. To solve this, we must break the motion into two distinct segments: the journey from the starting point $S$ to an intermediate point $Q$, and the subsequent journey from $Q$ to our target point $P$.

The beauty of this problem lies in the bridge between these two chapters: the velocity at point $Q$.

Let us focus on the segment from $Q$ to $P$. We are given that the distance $QP=400\text{ m}$ and the time taken is $t=4\text{ s}$. We also know the acceleration due to gravity is $g=10{\text{ m/s}}^2$.

We need to find the velocity at point $Q$, which we will call $v_Q$. For this segment, $v_Q$ acts as our initial velocity. Using the second equation of motion, $s=ut+\frac{1}{2}a{t}^2$, we can write:

Simplifying this, we get $400=4v_Q+80$. Subtracting $80$ from both sides gives $320=4v_Q$, which leads us to:

This velocity is the key that unlocks the first chapter.

Now, let us look at the first segment, from the starting point $S$ to point $Q$. We know the body started from rest at $S$, so the initial velocity $u_S=0$. We just calculated the final velocity at $Q$ to be $v_Q=80\text{ m/s}$.

Since we do not know the time for this segment, the third equation of motion, $v^2=u^2+2as$, is our most powerful tool. Substituting our values, we get:

Where $h_{SQ}$ is the height of the first segment. This simplifies to $6400=20h_{SQ}$. Solving for $h_{SQ}$, we find:

We have successfully navigated both chapters. The total height $H$ of the starting point $S$ above point $P$ is simply the sum of the two segments:

Substituting our values:

The body began its fall from a height of $720\text{ m}$ above point $P$. This problem teaches us that in multi-stage motion, the final state of one stage is the initial state of the next. By connecting these stages, we can solve even the most complex problems with elegance and precision.