Projectile Motion: Determining the Peak Height of a Cannonball

Question:

What is the formula to calculate the peak height a cannonball will rise when shot from level ground with an initial speed of 60 m/s at an angle of 28° above the horizontal?

Answer:

The formula to calculate the peak height a cannonball will rise to is determined by analyzing the vertical component of its motion. By breaking down the initial velocity into its vertical and horizontal components, we can calculate the peak height.

When a cannonball is shot from level ground with an initial speed of 60 m/s at an angle of 28° above the horizontal, we can determine the vertical component of the initial velocity using trigonometry. The vertical component of the initial velocity (Vy) can be calculated using the formula:

Vertical component of initial velocity (Vy) = Initial velocity (V) * sin(angle)

Given that the initial velocity is 60 m/s and the angle is 28°, we can plug in the values and calculate the vertical component of the initial velocity:

Vy = 60 m/s * sin(28°)

Using the approximation sin(28°) ≈ 0.4695, we find:

Vy ≈ 60 m/s * 0.4695 ≈ 28.17 m/s

Next, we can determine the time it takes for the cannonball to reach its peak height by dividing the vertical component of the initial velocity by the acceleration due to gravity (approximately 9.8 m/s²):

Time (t) = Vertical component of initial velocity (Vy) / Acceleration due to gravity (g)

Substituting the values, we get:

t = 28.17 m/s / 9.8 m/s² ≈ 2.87 s

Finally, to calculate the peak height, we multiply the vertical component of the initial velocity by the time:

Peak height = Vertical component of initial velocity (Vy) * Time (t)

Substituting the values, we find:

Peak height ≈ 28.17 m/s * 2.87 s ≈ 80.9 meters

This is the peak height that the cannonball will rise to in the absence of wind resistance.