Final answer:
To calculate how high the projectile reaches, the initial vertical velocity is found using the sine of the launch angle multiplied by the initial velocity. The maximum height is then found by rearranging the kinematic equation to solve for displacement using that vertical velocity and the acceleration due to gravity.
Step-by-step explanation:
To find how high a projectile launched at an angle of 22 degrees with an initial velocity of 830 m/s reaches, we first need to calculate the vertical component of the initial velocity. This can be done using trigonometric functions, specifically the sine function since we are dealing with the angle off the horizontal. The formula to calculate the vertical component of velocity (Vy) is Vy = V * sin(θ), where V is the initial velocity and θ is the launch angle.
The next step is to use the vertical component of the velocity to find the maximum height. At the maximum height, the vertical velocity will be 0 due to gravity. We can use the kinematic equation Vf^2 = Vi^2 + 2a*d, where Vf is the final velocity (which is 0 at the highest point), Vi is the initial velocity (Vy in this case), a is the acceleration due to gravity (which is -9.81 m/s^2, since it's in the opposite direction of the velocity), and d is the distance, which will be our maximum height.
Rearranging the formula to solve for d, we get d = (Vf^2 - Vi^2) / (2*a). Plugging in the values, we first calculate Vy = 830 m/s * sin(22°) and then use this Vy to find the height using the aforementioned kinematic equation.