Question

The velocity of a body falling under gravity is directly proportional to $g^a{h}^b$. If g and h are the acceleration due to gravity and height covered by the body, respectively, then determine the values of a and b.

• A. $\frac{1}{2}$ and $\frac{1}{2}$
• B. $-\frac{1}{2}$ and $-\frac{1}{2}$
• C. $\frac{1}{2}$ and $-\frac{1}{2}$
• D. $-\frac{1}{2}$ and $\frac{1}{2}$

Answer 1

Answer: (A)

$\frac{1}{2}$ and $\frac{1}{2}$

We know, one of the fundamental equation of motion is
$v^2-u^2=2as$
Where, v is Final Velocity
u is Initial Velocity
a is Acceleration
t is time taken.
In case of motion under gravity, the acceleration a and the distance s in the above equations are replaced by gravity g and the height of the object.
Thus the acceleration due to gravity equation is,
$v^2-u^2=2gh$
Where,
h is Height from ground level and
g is acceleration due to gravity.
For a body falling under gravity initially at rest i.e. u = 0, hence
$v^2=2gh$
$v=\sqrt{2}gh$
$\Rightarrow v\propto \sqrt{g}h$
$\Rightarrow v\propto g^{\frac{1}{2}}{h}^{\frac{1}{2}}$