In 1939 or 1940, Emanuel Zacchini took his human-cannonball act to an extreme: After being shot from a cannon, he soared over three Ferris wheels and into a net (see the figure). Assume that he is launched with a speed of 29 m/s and at an angle of 59°. (a) Treating him as a particle, calculate his clearance over the first wheel. (b) If he reached maximum height over the middle wheel, by how much did he clear it? (c) How far from the cannon should the net's center have been positioned (neglect air drag)?
Give me a detailed answer
To solve this problem, we can use the equations of motion to analyze the projectile motion of Emanuel Zacchini.
Given:
Initial speed (vi) = 29 m/s
Launch angle (θ) = 59°
(a) Clearance over the first wheel:
To find the clearance over the first wheel, we need to calculate the maximum height reached by the projectile. The clearance over the first wheel will be equal to this maximum height.
The maximum height (h) can be found using the equation:
h = (vi^2 * sin^2(θ)) / (2 * g)
Where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Plugging in the values:
h = (29^2 * sin^2(59°)) / (2 * 9.8)
h ≈ 44.19 m
So, the clearance over the first wheel is approximately 44.19 meters.
(b) Clearance over the middle wheel:
To find the clearance over the middle wheel, we can use the equation for the vertical displacement at maximum height (ymax):
ymax = (vi^2 * sin^2(θ)) / (2 * g)
Since the projectile reaches maximum height over the middle wheel, the clearance over the middle wheel will be twice this value.
Clearance over the middle wheel = 2 * ymax
Plugging in the values:
ymax = (29^2 * sin^2(59°)) / (2 * 9.8)
ymax ≈ 44.19 m
Clearance over the middle wheel = 2 * 44.19
Clearance over the middle wheel ≈ 88.38 m
So, he cleared the middle wheel by approximately 88.38 meters.
(c) Position of the net's center:
To find the position of the net's center, we need to determine the horizontal displacement (range) of the projectile. The range (R) can be calculated using the equation:
R = (vi^2 * sin(2θ)) / g
Plugging in the values:
R = (29^2 * sin(2 * 59°)) / 9.8
R ≈ 46.61 m
The net's center should be positioned at a horizontal distance of approximately 46.61 meters from the cannon.
Please note that this calculation neglects air drag. In reality, air drag may have some effect on the actual clearance and range.
To solve this problem, we can use the equations of projectile motion. Here's how you can tackle each sub-question:
(a) To calculate the clearance over the first wheel, we need to find the maximum height reached by Emanuel Zacchini. The vertical motion can be analyzed independently of the horizontal motion.
Using the equation for the maximum height reached in projectile motion, we have:
h_max = (V_initial^2 * sin^2(theta)) / (2 * g)
where:
V_initial = initial velocity = 29 m/s
theta = launch angle = 59°
g = acceleration due to gravity ≈ 9.8 m/s^2
Substituting the given values into the equation, we get:
h_max = (29^2 * sin^2(59°)) / (2 * 9.8)
Calculating this expression will give you the maximum height reached by Emanuel Zacchini. The clearance over the first wheel will be equal to this height.
(b) To calculate how much Emmanuel Zacchini cleared over the middle wheel, we can use the same equation for the maximum height. Since the middle wheel is at the maximum height, the clearance over it will be equal to the maximum height.
(c) To determine how far from the cannon the net's center should be positioned, we need to find the horizontal range traveled by Emanuel Zacchini. We can use the equation for the horizontal range:
R = (V_initial^2 * sin(2 * theta)) / g
where:
R = horizontal range
V_initial = initial velocity = 29 m/s
theta = launch angle = 59°
g = acceleration due to gravity ≈ 9.8 m/s^2
Substituting the given values into the equation, we get:
R = (29^2 * sin(2 * 59°)) / 9.8
Calculating this expression will give you the horizontal range traveled by Emanuel Zacchini. The net's center should be positioned at this horizontal distance from the cannon.
Remember to convert angles to radians when using trigonometric functions. (sin(59°) should be sin(59° * pi/180))