To find the initial speed of the cannonball, we need to use the concept of vector addition. The initial horizontal and vertical velocities of the cannonball can be combined using the Pythagorean theorem.
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, the initial horizontal velocity (33.0 m/s) and the initial vertical velocity (25.0 m/s) are the two sides of the triangle, and the initial speed we are trying to find is the hypotenuse.
Using the Pythagorean theorem equation:
initial speed^2 = (initial horizontal velocity)^2 + (initial vertical velocity)^2
Plugging in the values:
initial speed^2 = (33.0 m/s)^2 + (25.0 m/s)^2
Simplifying:
initial speed^2 = 1089 m^2/s^2 + 625 m^2/s^2
initial speed^2 = 1714 m^2/s^2
To get the initial speed, we need to take the square root of both sides of the equation:
initial speed = √(1714 m^2/s^2)
Using a calculator, we find that the initial speed of the cannonball is approximately 41.40 m/s.