A golfer imparts a speed of 25.8 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?
See the answers below.
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To answer both parts of the question, we can use the equations of motion for the ball's projectile motion. We'll assume that there is no air resistance and neglect the effects of gravity over the horizontal distance covered by the ball.
Let's start by solving part (a):
(a) How much time does the ball spend in the air?
The time of flight (t) can be calculated using the formula:
t = (2 * V₀ * sin(θ)) / g,
where:
V₀ is the initial velocity (speed) of the ball,
θ is the angle of projection, and
g is the acceleration due to gravity.
In this case, the ball is hit at an angle of 45 degrees (θ = 45°) because the golfer wants to maximize the distance without considering the elevation of the green. The initial velocity is given as 25.8 m/s (V₀ = 25.8 m/s). The acceleration due to gravity is approximately 9.8 m/s² (g = 9.8 m/s²).
Now we can substitute the values into the formula:
t = (2 * 25.8 * sin(45°)) / 9.8.
Calculating this equation:
t = (2 * 25.8 * 0.7071) / 9.8,
t = 2 * 0.7071 * 2.6327,
t ≈ 2.95 seconds.
Therefore, the ball spends approximately 2.95 seconds in the air.
Moving on to part (b):
(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 the ball does not roll when it hits the green, we are interested in the horizontal distance covered by the ball. We can calculate this using the formula:
R = (V₀² * sin(2θ)) / g,
where R is the range or horizontal distance traveled.
The angle of projection remains the same (θ = 45°), and the initial velocity is still 25.8 m/s.
Now, substitute these values into the formula:
R = (25.8² * sin(90°)) / 9.8,
Calculating this equation:
R = 668.52 / 9.8,
R ≈ 68.2 meters.
Therefore, the longest "hole in one" that the golfer can make, assuming the ball doesn't roll when it hits the green, is approximately 68.2 meters.