Two bullets are fired simultaneously, horizontally and with different speeds from the same place. Which bullet will hit the ground first

To determine which bullet hits the ground first, we need to analyze the motion of the bullets. Each bullet undergoes two-dimensional projectile motion, which can be separated into independent horizontal and vertical components.

Let the height from which both bullets are fired be $h$.
Since the bullets are fired horizontally, their initial vertical velocity ($u_y$) is zero for both bullets:
$u_y=0$

The only force acting on the bullets vertically after they are fired is gravity. Therefore, both bullets experience the same downward vertical acceleration, which is the acceleration due to gravity ($g$):
$a_y=g$

We can use the second equation of motion for the vertical direction to find the time ($t$) it takes for a bullet to hit the ground:
$h=u_{y}t+\frac{1}{2}{a}_y{t}^2$

Substituting $u_y=0$ and $a_y=g$ into the equation:
$h=0\cdot t+\frac{1}{2}g{t}^2$
$h=\frac{1}{2}g{t}^2$

Solving for the time $t$ gives:
$t^2=\frac{2h}{g}$
$t=\sqrt{\frac{2h}{g}}$

This formula shows that the time of flight depends only on the vertical height ($h$) and the acceleration due to gravity ($g$). It is completely independent of the horizontal speed of the bullets as well as their masses.

Since both bullets are fired from the same height ($h$) and experience the same gravitational acceleration ($g$), they will take the exact same amount of time to reach the ground. Thus, both will reach the ground simultaneously.