2) A ball is hit at an angle of 36 degrees to the horizontal from 14 feet above the ground. If it hits the ground 26 feet away (in horizontal distance), what was the initial velocity? Assume there is no air resistance.

To find the initial velocity of the ball, you can use the horizontal and vertical components of its motion.

Let's break down the problem into two parts: the horizontal component and the vertical component.

1. Horizontal Component:
Since there is no air resistance, the horizontal component of the ball's motion remains constant throughout its trajectory. In this case, the horizontal distance traveled by the ball is given as 26 feet.

The formula for horizontal distance is:
Horizontal Distance = (Initial Velocity) * (Time)

However, we don't know the time it takes for the ball to travel 26 feet. To find the time, we can use the vertical motion of the ball.

2. Vertical Component:
The vertical component of the ball's motion is influenced by the force of gravity. We can use the following equations to solve for the initial velocity and time.

a. Vertical Distance Formula:
Vertical Distance = (Initial Vertical Velocity) * (Time) + (0.5) * (Gravity Acceleration) * (Time^2)

b. Vertical Velocity Formula:
Final Vertical Velocity = Initial Vertical Velocity + (Gravity Acceleration) * (Time)

The ball is initially hit at an angle of 36 degrees to the horizontal from 14 feet above the ground, so we can calculate the initial vertical velocity using trigonometry:

Initial Vertical Velocity = Initial Velocity * sin(36)

Now, let's find the time it takes for the ball to hit the ground. At the moment the ball hits the ground, its vertical distance will be zero.

Using equation a from the above vertical component, we have:
0 = (Initial Vertical Velocity) * (Time) + (0.5) * (Gravity Acceleration) * (Time^2)

Substituting the value of the initial vertical velocity, we get:
0 = (Initial Velocity * sin(36)) * (Time) - (0.5) * (Gravity Acceleration) * (Time^2)

Since we now have an equation with one unknown (time), we can solve for time using the quadratic formula. Ignore the negative value for time, as we are only interested in the positive time it takes for the ball to hit the ground.

Once we find the time, we can substitute it into the formula for horizontal distance to solve for the initial velocity:

26 = (Initial Velocity) * (Time)

Now you can solve for the initial velocity using the time you found earlier.

Note: The value of the gravity acceleration should be consistent with your units of measurement (e.g., feet per second squared if using feet for distance and seconds for time).