Question

A stone is dropped from a height of $9$ meters above the ground. If the height functions can be modelled by the equation $h(t) = a - {t^2}$, where $t$ is the time in seconds, $h$ is the height in meters, and $a$ is the initial height, how many seconds does it take for the stone to hit the ground?

Answer 1

Hint: In the solution, first we have to find the initial position of the stone. Since the height of the stone is given, so by putting $t = 0\,{\rm{s}}$ in the function we have to calculate the value of $a$. Finally, we have to substitute $h(t) = 0$ and the value of $a$ and solve for $t$.

Complete step by step answer:
Objects which are said to be experiencing free fall, do not experience a significant force of air resistance; they come under the single control of gravity. Under these conditions, all objects, regardless of their mass, will fall with the same acceleration rate. A free falling object is an object that comes under the sole gravitational effect. Anybody which only the force of gravity acts on is said to be in a state of free fall.
Given, the initial height at which the stone is present i.e. at $\left( {t = 0\,{\rm{s}}} \right)$ is $9$ meters.
This can also be depicted by $h\left( 0 \right)$.
Given function of height is
$h(t) = a - {t^2}$
By putting $t = 0\,{\rm{s}}$, in the above function, we obtain as follows:
$\begin{array}{c} \Rightarrow h\left( 0 \right) = a - {0^2}\\ \Rightarrow 9 = a\\ \Rightarrow a = 9\end{array}$
The initial position of the stone comes out to be at a height of $9$ meters from the ground.
After the stone is dropped, when it hits the ground the height becomes zero. This is because you take the relative position of the stone with respect to the ground.
Hence, $h(t) = 0$.
So, we can write:
$\begin{array}{c} \Rightarrow h(t) = a - {t^2}\\ \Rightarrow 0 = 9 - {t^2}\\ \Rightarrow {t^2} = 9\\ \Rightarrow t = 3\,{\rm{s}}\end{array}$

Hence, the time required to hit the ground is $3$ seconds.

Note: In this problem you are asked to find the time taken by the stone to hit the ground. First of all, you need to find the initial position of the stone above the ground. After you have got the initial position of the stone, then put $h(t) = 0$ and $a = 9$ in the function to find the time in seconds.