Explanation:

To understand why the maximum distance through which a stone can be thrown horizontally is 2h, we need to consider the projectile motion of the stone.

Projectile Motion:
When an object is thrown into the air, it follows a curved path known as projectile motion. This motion can be divided into two independent components: horizontal motion and vertical motion.

Horizontal Motion:
The horizontal motion of the stone remains constant throughout its trajectory. This means that the stone will travel the same horizontal distance regardless of the time it takes to reach the maximum height.

Vertical Motion:
The vertical motion of the stone is affected by gravity. As the stone is thrown vertically, it will reach a maximum height and then fall back to the ground. The time taken to reach the maximum height is the same as the time taken to fall back to the ground.

Maximum Height:
At the maximum height, the vertical velocity of the stone becomes zero. This means that the stone has reached its highest point and is about to fall back down.

Time of Flight:
The total time taken for the stone to reach the maximum height and fall back to the ground is known as the time of flight. It can be calculated using the equation:

t = 2 * (v₀/g)

Where:
t = Time of flight
v₀ = Initial vertical velocity
g = Acceleration due to gravity

Horizontal Distance:
The horizontal distance traveled by the stone can be calculated using the equation:

d = v₀ * t

Where:
d = Horizontal distance
v₀ = Initial horizontal velocity
t = Time of flight

Since the initial horizontal velocity is constant throughout the trajectory, we can substitute it with the horizontal component of the initial velocity (v₀x).

Horizontal and Vertical Components of Velocity:
The initial velocity of the stone can be resolved into horizontal and vertical components using trigonometry:

v₀x = v₀ * cos(θ)
v₀y = v₀ * sin(θ)

Where:
v₀x = Horizontal component of initial velocity
v₀y = Vertical component of initial velocity
θ = Angle of projection

Maximum Distance:
In the given problem, the stone is thrown to a maximum height of h meters vertically. This means that the vertical component of the initial velocity (v₀y) is h m/s.

Using the equation for time of flight, we can calculate the total time taken for the stone to reach the maximum height and fall back to the ground:

t = 2 * (v₀y/g)
t = 2 * (h/g)

Substituting the time of flight into the equation for horizontal distance, we get:

d = v₀x * t
d = v₀ * cos(θ) * t
d = v₀ * cos(θ) * 2 * (h/g)

Since the stone is thrown horizontally, the angle of projection (θ) is 0 degrees. This means that the cosine of 0 degrees is 1:

d = v₀ * 1 * 2 * (h/g)
d = 2 * v₀ * (h/g)

Since the initial horizontal velocity (v₀x) is equal to the initial velocity (v₀), we can substitute it into the equation:

d = 2 * v₀ * (h/g)
d