Question

A stone is thrown vertically upwards from the top of a tower 64 m high according to the law of motion $s = 48\,t - 16\,{t^2}$. The greatest height attained by the stone above ground is
A.36 m
B.32 m
C.100 m
D.64 m

Answer 1

Hint : Here in this question, we need to find the greatest height attained by the stone ground by using a given equation of law of motion $s = 48\,t - 16\,{t^2}$. For this, first we need to find the time $t$ by differentiating the given equation then substitute the value of $t$ on the equation and on further simplification to get the required solution.

Complete step-by-step answer :
Consider the question,
A stone is thrown vertically upwards from the top of a tower is 64 m.
Given, the equation of law of motion
$s = 48\,t - 16\,{t^2}$ -----(1)
Here, s is a dependent variable and t is an independent variable.
Now we have to differentiate the equation (1) with respect to t.
$\Rightarrow \,\dfrac{{ds}}{{dt}} = \dfrac{d}{{dt}}\left( {48\,t - 16\,{t^2}} \right)$
$\Rightarrow \,\dfrac{{ds}}{{dt}} = 48\,\dfrac{d}{{dt}}\left( t \right) - 16\dfrac{d}{{dt}}\left( {{t^2}} \right)$ ----(2)
Using the standard differentiated formula $\dfrac{d}{{dx}}\left( {{x^n}} \right) = n{x^{n - 1}}$, then equation (2) becomes
$\Rightarrow \,\dfrac{{ds}}{{dt}} = 48\,\left( 1 \right) - 16\left( {2t} \right)$
$\Rightarrow \,\dfrac{{ds}}{{dt}} = 48\, - 32\,t$ ------(3)
For greatest height velocity is equal to zero i.e., $v = \dfrac{{ds}}{{dt}} = 0$, then the equation (3) becomes
$\Rightarrow \,0 = 48\, - 32\,\,t$
Add 32t on both side
$\Rightarrow \,32\,t = 48\,$
Divide 32 on both side
$\Rightarrow \,\,t = \dfrac{{48}}{{32}}\,$
Divide both numerator and denominator by 16, then we get
$\Rightarrow \,\,t = \dfrac{3}{2}\,$ ------(4)
Substitute ‘$t$’ value in equation (1)
$\Rightarrow s = 48\,\left( {\dfrac{3}{2}} \right) - 16\,{\left( {\dfrac{3}{2}} \right)^2}$
$\Rightarrow s = 24\,\left( 3 \right) - 16\,\left( {\dfrac{9}{4}} \right)$
$\Rightarrow s = 24\,\left( 3 \right) - 4\,\left( 9 \right)$
$\Rightarrow s = 72 - 36$
$\therefore s = 36\,\,m$
The greatest height attained by the stone above the top of the tower is 36 m
Hence, The total height or the greatest height attained by the stone above ground is:
$\Rightarrow \,64 + 36$
$\therefore \,100\,m$
Therefore, option (C) is the correct answer.
So, the correct answer is “Option C”.

Note : The velocity $v$ is defined as the rate of change of displacement $s$ i.e., $\dfrac{{ds}}{{dt}}$, the velocity at the greatest height or maximum height is always equal to 0. When differentiating the function or term the student must recognize the dependent and independent variable then differentiate the dependent variable with respect to independent variable and should remember the standard differentiation formulas.