**Projectile Motion and Range**

Projectile motion refers to the motion of an object that is launched into the air and is subject only to the force of gravity and air resistance. When a projectile is thrown with an initial velocity (vo) at an angle (α) to the horizontal, it follows a curved path known as a parabola.

The range (R) of a projectile is the horizontal distance traveled by the object before it hits the ground or any other target. It can be calculated using the formula:

R = (vo^2 * sin(2α)) / g

where g is the acceleration due to gravity.

**Striking a Vertical Wall**

Let's consider a projectile thrown at an angle α, with an initial velocity vo. We are given that it strikes a vertical wall at a distance R/2 from the point of projection.

To determine the speed at which it strikes the wall, we need to find the horizontal component of the velocity (v_x) at the point of impact. This can be calculated by multiplying the initial velocity (vo) by the cosine of the launch angle (α):

v_x = vo * cos(α)

Since the projectile strikes the wall at a distance R/2 from the point of projection, we know that the time of flight (t) is half the total time it takes to complete the trajectory.

Using the equation of motion for the horizontal direction:

R/2 = v_x * t

Substituting the expression for v_x and rearranging the equation:

R/2 = (vo * cos(α)) * (R / (vo * sin(α) * g))

Simplifying the equation:

1/2 = cos(α) / (sin(α) * g)

Solving for cos(α):

cos(α) = (1/2) * (sin(α) * g)

cos(α) = (1/2) * (2 * sin(α) * cos(α) * g)

cos(α) = sin(α) * cos(α) * g

Dividing both sides by (sin(α) * cos(α)):

1 = g

Therefore, the speed at which the projectile strikes the wall is equal to the acceleration due to gravity (g).

At a horizontal distance of R/2 from the point of projection means at that distance when the body attains maximum height ( due to symmetry of the projectile) so the wall's height= h Max. and at that instant V = Vo cos alpha