A golfer imparts a speed of 32.1 m/s to a ball, and it travels the maximum possible distance before landing on the green. The tee and the green are at the same elevation.

(a) How much time does the ball spend in the air?
(b) What is the longest "hole in one" that the golfer can make, if the ball does not roll when it hits the green?

(a) To find the time the ball spends in the air, we can use the equation for the time of flight of a projectile:

t = (2 * V * sin(theta)) / g

Where:
t is the time of flight
V is the initial velocity (speed) of the ball
theta is the launch angle
g is the acceleration due to gravity (approximately 9.8 m/s^2)

Since the ball is traveling the maximum possible distance, the launch angle would be 45 degrees, as this will maximize the horizontal distance traveled.

So, plugging in the values:
V = 32.1 m/s
theta = 45 degrees
g = 9.8 m/s^2

t = (2 * 32.1 * sin(45)) / 9.8

Using a calculator, this simplifies to:
t = 4.148 seconds (rounded to three decimal places)

Therefore, the ball spends approximately 4.148 seconds in the air.

(b) To find the longest "hole in one" distance, we need to calculate the horizontal distance traveled by the ball.

The horizontal distance traveled can be calculated using the following equation:
d = V * cos(theta) * t

Where:
d is the horizontal distance
V is the initial velocity (speed) of the ball
theta is the launch angle
t is the time of flight

Using the same values as before:
V = 32.1 m/s
theta = 45 degrees
t = 4.148 seconds

d = 32.1 * cos(45) * 4.148

Using a calculator, this simplifies to:
d = 112.472 meters (rounded to three decimal places)

Therefore, the longest "hole in one" distance the golfer can make, if the ball does not roll when it hits the green, is approximately 112.472 meters.

To find the answer to this question, we can use the equations of motion for a projectile in free-fall motion. Let's break it down step by step:

(a) How much time does the ball spend in the air?

In projectile motion, the vertical motion and horizontal motion of the ball are independent of each other. Since the tee and the green are at the same elevation, the vertical component of the motion will follow a parabolic path, and the time of flight will be the same as in the horizontal motion.

The time of flight can be determined using the equation:

t = 2 * (v₀y) / g

Where:
t is the time of flight
v₀y is the initial vertical velocity component
g is the acceleration due to gravity (approximately 9.8 m/s²)

In this case, the ball is initially hit at an angle such that the entire initial velocity contributes to the horizontal component (v₀x) and none of it contributes to the vertical component (v₀y).

Therefore, v₀y = 0 and the time of flight (t) = 0.

(b) What is the longest "hole in one" that the golfer can make, if the ball does not roll when it hits the green?

Since there is no horizontal acceleration acting on the ball once it is in the air, the horizontal distance traveled (d) can be calculated using the equation:

d = v₀x * t

In this case, the initial horizontal velocity component (v₀x) is equal to the total initial velocity (32.1 m/s), and t = 0.

Therefore, the horizontal distance traveled (d) is also zero.

In summary, the ball spends zero time in the air and travels zero distance horizontally before landing on the green.