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 equations of motion for projectile motion. The key information we have is the initial speed of the projectile (150 meters per second) and the launch angle (45 degrees above the horizontal).
The first step is to break down the initial velocity of the projectile into its vertical and horizontal components. Given that the launch angle is 45 degrees, we can use trigonometry to determine the initial vertical and horizontal velocities.
The initial vertical velocity (Vy) can be found using the formula: Vy = V * sin(θ), where V is the initial speed and θ is the launch angle. Substituting the values, Vy = 150 * sin(45).
The initial horizontal velocity (Vx) can be found using the formula: Vx = V * cos(θ), where V is the initial speed and θ is the launch angle. Substituting the values, Vx = 150 * cos(45).
Now, we know that the time taken to reach the maximum height is half the total time of flight. Assuming the projectile lands at the same level it was launched from, we can calculate the total time of flight using the formula for projectile motion: T = (2 * Vy) / g, where g is the acceleration due to gravity (approximately 9.8 meters per second squared).
Since the time for reaching the maximum height is half of T, we can calculate it as Tmax = T / 2.
Next, we can find the maximum height (Hmax) reached by the projectile. Using the formula for vertical displacement: Hmax = Vy * Tmax - (1/2) * g * Tmax^2.
Substituting the calculated values, we can find the value of Hmax.
Now, let's calculate the maximum height reached by the projectile:
Vy = 150 * sin(45) = 150 * 0.707 = 106.1 meters per second
Vx = 150 * cos(45) = 150 * 0.707 = 106.1 meters per second
T = (2 * Vy) / g = (2 * 106.1) / 9.8 = 21.6 seconds
Tmax = T / 2 = 21.6 / 2 = 10.8 seconds
Hmax = 106.1 * 10.8 - (1/2) * 9.8 * (10.8^2) = 574.8 - 572.9 = 1.9 meters
Therefore, the maximum height reached by the projectile is approximately 1.9 meters.