Question

Use the model for projectile motion, assuming there is no air resistance. Determine the maximum height and range of a projectile fired at a height of 3 feet above the ground with an initial velocity of 900 feet per second and at an angle of \(45^{\circ}\) above the horizontal.

Short answer

The maximum height of the projectile is approximately 10107.2 feet and the range it covers is approximately 25264.3 feet.

Step-by-step solution

Calculating maximum height

First, calculate the maximum height by filling the given input into the suitable formula for maximum height:\( h = \frac{900^2 \cdot \sin^2(45)}{2 * 32.2} + 3 \)

Simplifying maximum height

Simplify the above equation to compute maximum height.First compute the sine value, squared it and then multiply it by the square of the initial velocity. Lastly, divide by 2 times the acceleration and add with the initial height.

Calculating range

Now calculate the range of the projectile using the formula:\( R = \frac{900^2 \cdot \sin(2 * 45)}{32.2} \)

Simplifying range

Simplify the formula by first calculating the sine of twice the angle and then multiply it with the square of the initial velocity. Finally, divide the entire expression by the acceleration due to gravity which should give the range.