A golf ball has been hit off of the tee at an angle of elevation of 30 degrees and an initial velocity of 128 ft/sec.Using the helpful parametric models for projectile motion answer the following questions.a.) How long is the ball in the air (hang time)? b.) What is the maximum height of the ball? c.) How far, horizontally, does the ball travel in the air? (round to the nearest foot)
Question
A golf ball has been hit off of the tee at an angle of elevation of 30 degrees and an initial velocity of 128 ft/sec.Using the helpful parametric models for projectile motion answer the following questions.a.) How long is the ball in the air (hang time)? b.) What is the maximum height of the ball? c.) How far, horizontally, does the ball travel in the air? (round to the nearest foot)
Solution
Sure, let's solve this step by step.
a.) How long is the ball in the air (hang time)?
The time a projectile is in the air is determined by its initial velocity and the acceleration due to gravity. The formula for time of flight is:
t = (2 * v * sin(θ)) / g
where: v = initial velocity = 128 ft/sec θ = launch angle = 30 degrees g = acceleration due to gravity = 32.2 ft/sec²
First, convert the launch angle to radians:
θ = 30 * (π / 180) = π / 6 rad
Then, plug the values into the formula:
t = (2 * 128 * sin(π / 6)) / 32.2 ≈ 8 seconds
So, the ball is in the air for approximately 8 seconds.
b.) What is the maximum height of the ball?
The maximum height a projectile reaches is determined by its initial velocity, launch angle, and the acceleration due to gravity. The formula for maximum height is:
h = (v² * sin²(θ)) / (2 * g)
Plug the values into the formula:
h = (128² * sin²(π / 6)) / (2 * 32.2) ≈ 64 feet
So, the maximum height of the ball is approximately 64 feet.
c.) How far, horizontally, does the ball travel in the air?
The horizontal distance a projectile travels is determined by its initial velocity, launch angle, and the time it's in the air. The formula for horizontal distance is:
d = v * cos(θ) * t
Plug the values into the formula:
d = 128 * cos(π / 6) * 8 ≈ 892 feet
So, the ball travels approximately 892 feet horizontally.
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