A ball A is projected from the ground such that its horizontal range is maximum. Another ball B is dropped from a height equal to the maximum range of A. The ratio of the time of flight of A to the time of fall of B is 

To find the ratio of the time of flight of ball A to the time of fall of ball B, let's consider the horizontal motion of ball A and the vertical motion of ball B separately.

For ball A:
The horizontal range of a projectile is given by the formula:

$R=\frac{v^2\sin \left(2\theta \right)}{g}$

where:
R = horizontal range
v = initial velocity
θ = launch angle
g = acceleration due to gravity

To maximize the horizontal range, we need to choose the launch angle θ such that $\sin \left(2\theta \right)=1$, which means $2\theta =\frac{\pi }{2}$ or $\theta =\frac{\pi }{4}$.

Now, let's consider the vertical motion of ball B. When an object is dropped from a height h, the time of fall can be found using the formula:

$h=\frac{1}{2}g{t}^2$

where:
h = height
g = acceleration due to gravity
t = time of fall

Since ball B is dropped from a height equal to the maximum range of ball A, the height h is equal to the maximum range R of ball A.

Now, let's calculate the time of flight of ball A and the time of fall of ball B.

For ball A:
The time of flight can be calculated using the formula:

$T_A=\frac{2v\sin \left(\theta \right)}{g}$

Substituting the value of $\theta = \frac{\pi }{4}$:

$T_A=\frac{2v\sin \left(\frac{\pi }{4}\right)}{g}=\frac{2v}{g}\left(\frac{1}{\sqrt{2}}\right)=\frac{\sqrt{2}v}{g}$

For ball B:
Since ball B is dropped from a height equal to the maximum range of ball A, the time of fall is the same as the time taken for ball A to reach its maximum range. Therefore, the time of fall of ball B is also given by:

$T_B=\frac{\sqrt{2}v}{g}$

Now, let's calculate the ratio of the time of flight of ball A to the time of fall of ball B:

$\frac{T_A}{T_B}=\frac{\frac{\sqrt{2}v}{g}}{\frac{\sqrt{2}v}{g}}=\frac{1}{1}$

Therefore, the ratio of the time of flight of ball A to the time of fall of ball B is 1:1. 

Hence, the correct answer is option c) 1:1.