Alexis Farias realizes her dream to play golf on the moon. She hits a golf ball with an initial velocity of 23.5m/s at an angle of 35.0 degrees above the horizontal. The acceleration due to gravity on the moon is 1.67m/s^2.

A). How long is the golf ball in the air?
B). How far does Alexis hit the golf ball?
C). What is the maximum height of the ball?

To solve this problem, we need to break it down into smaller parts using projectile motion equations. Projectile motion refers to the motion of objects that are launched into the air and are subject only to the force of gravity and air resistance (which we can ignore in this case since we are on the moon).

To solve for the golf ball's motion, we can use the following equations:
1. The horizontal component of the velocity: vx = v * cos(theta)
2. The vertical component of the velocity: vy = v * sin(theta)
3. The time the ball is in the air: t = 2 * vy / g
4. The horizontal distance the golf ball travels: dx = vx * t
5. The maximum height the ball reaches: h = (vy^2) / (2 * g)

Given that:
- Initial velocity (v) = 23.5 m/s
- Launch angle (theta) = 35.0 degrees
- Acceleration due to gravity on the moon (g) = 1.67 m/s^2

Let's solve the problem step by step:

A). How long is the golf ball in the air?
To find the time the ball is in the air (t), we can use equation 3:
t = 2 * vy / g

First, calculate the vertical component of velocity (vy):
vy = v * sin(theta)

So, vy = 23.5 * sin(35.0)

Now, substitute the values in the equation:
t = 2 * (23.5 * sin(35.0)) / 1.67

Simplify the equation to find the time:
t = 2 * (23.5 * 0.5745) / 1.67
t = 2 * 13.49825 / 1.67
t ≈ 25.60 seconds

Therefore, the golf ball is in the air for approximately 25.60 seconds.

B). How far does Alexis hit the golf ball?
To find the horizontal distance the ball travels (dx), we can use equation 4:
dx = vx * t

First, calculate the horizontal component of velocity (vx):
vx = v * cos(theta)

So, vx = 23.5 * cos(35.0)

Now, substitute the values in the equation:
dx = (23.5 * cos(35.0)) * 25.60

Simplify the equation to find the horizontal distance:
dx ≈ (23.5 * 0.8192) * 25.60
dx ≈ 376.12 m

Therefore, Alexis hits the golf ball approximately 376.12 meters.

C). What is the maximum height of the ball?
To find the maximum height (h), we can use equation 5:
h = (vy^2) / (2 * g)

First, calculate the vertical component of velocity (vy):
vy = v * sin(theta)

So, vy = 23.5 * sin(35.0)

Now, substitute the values in the equation:
h = (23.5 * sin(35.0))^2 / (2 * 1.67)

Simplify the equation to find the maximum height:
h ≈ (23.5 * 0.5745)^2 / (2 * 1.67)
h ≈ 3.345 m

Therefore, the maximum height of the ball is approximately 3.345 meters.