A projectile is fired from the surface of the Earth with a speed of 150 meters per second at an angle of 45 degrees above the horizontal. If the ground is level, what is the maximum height reached by the projectile?

To find the maximum height reached by the projectile, we can use the principles of projectile motion.

The motion of the projectile can be divided into two components: horizontal and vertical.

First, let's calculate the time it takes for the projectile to reach its maximum height. The time of flight (t) for the whole trajectory can be determined using the vertical component of the velocity.

Using the equation:
t = (2 * V * sin θ) / g,

where:
V = initial velocity of the projectile (150 m/s),
θ = angle of projection (45 degrees),
g = acceleration due to gravity (9.8 m/s²),

t = (2 * 150 * sin 45) / 9.8 = 30 / 9.8 ≈ 3.06 seconds.

Next, we can calculate the maximum height (h) reached by the projectile using the height formula:
h = (V * sin θ)² / (2 * g).

Plugging in the values:
h = (150 * sin 45)² / (2 * 9.8) = (150 * 0.7071)² / 19.6 = 88.575 / 19.6 ≈ 4.51 meters.

Therefore, the maximum height reached by the projectile is approximately 4.51 meters.