15. During the opening kickoff of a college football game, the kicker kicks a football with an initial velocity of 27.5 m/s at an angle of 41° above horizontal.
I. What is the time of flight for the ball?
II. How far does it travel before hitting the ground?
III. What is the maximum height the football reaches?
first half of flight:
u = 27.5 cos 41 = 20.75 m/s forever
Vi = 27.5 sin 41 = 18 m/s initial speed up
v = 18 - 9.81 t
at top v = 0
9.81 t = 18
t = 1.84 seconds upward
2 t = 3.68 seconds in air
3.68 seconds* 20.75 m/s = 76.3 meters range
h = 18 (1.84) -4.9(1.84)^2
bro im bad at this
To solve for the time of flight, distance traveled, and maximum height reached by the football during the opening kickoff, we can use the equations of projectile motion.
I. Time of flight:
The time of flight (t) is the total time in the air. We can calculate it using the equation:
t = (2 * v * sin(θ)) / g
Where:
- v is the initial velocity (27.5 m/s)
- θ is the angle of projection (41°)
- g is the acceleration due to gravity (approximately 9.8 m/s²)
Substituting the given values into the equation:
t = (2 * 27.5 * sin(41°)) / 9.8
Calculating this expression gives:
t ≈ 3.55 seconds
Therefore, the time of flight for the ball is approximately 3.55 seconds.
II. Distance traveled:
To find out how far the ball travels horizontally before hitting the ground, we can use the equation:
horizontal distance (d) = v * cos(θ) * t
Where:
- v is the initial velocity (27.5 m/s)
- θ is the angle of projection (41°)
- t is the time of flight (3.55 seconds)
Substituting the given values into the equation:
d = 27.5 * cos(41°) * 3.55
Calculating this expression gives:
d ≈ 70.28 meters
Therefore, the ball travels approximately 70.28 meters before hitting the ground.
III. Maximum height:
The maximum height (h) reached by the ball can be calculated using the equation:
h = (v² * sin²(θ)) / (2 * g)
Where:
- v is the initial velocity (27.5 m/s)
- θ is the angle of projection (41°)
- g is the acceleration due to gravity (approximately 9.8 m/s²)
Substituting the given values into the equation:
h = (27.5² * sin²(41°)) / (2 * 9.8)
Calculating this expression gives:
h ≈ 14.06 meters
Therefore, the maximum height reached by the football is approximately 14.06 meters.
To find the answers to the given questions, we can use the laws of projectile motion. Let's break down each question and solve them step by step:
I. What is the time of flight for the ball?
The time of flight refers to the total time the ball remains in the air. To find it, we will use the vertical component of the initial velocity.
1. We can find the vertical component of the initial velocity (Vyi) using trigonometry:
Vyi = V * sin(θ)
Vyi = 27.5 m/s * sin(41°)
Vyi ≈ 17.72 m/s
2. The time of flight can be calculated using the equation:
t = (2 * Vyi) / g
where g is the acceleration due to gravity (approximately 9.8 m/s^2)
t = (2 * 17.72 m/s) / 9.8 m/s^2
t ≈ 3.63 seconds
Therefore, the time of flight for the ball is approximately 3.63 seconds.
II. How far does it travel before hitting the ground?
To find the horizontal range, we need to calculate the horizontal component of the initial velocity.
1. We can find the horizontal component of the initial velocity (Vxi) using trigonometry:
Vxi = V * cos(θ)
Vxi = 27.5 m/s * cos(41°)
Vxi ≈ 20.85 m/s
2. The horizontal range (R) can be calculated using the equation:
R = Vxi * t
R = 20.85 m/s * 3.63 s
R ≈ 75.61 meters
Therefore, the ball travels approximately 75.61 meters before hitting the ground.
III. What is the maximum height the football reaches?
To find the maximum height, we need to calculate the vertical displacement of the ball.
1. The maximum height occurs when the ball reaches its peak (at which the vertical velocity becomes zero).
Using the equation:
Vf = Vi + (-g * t)
where Vf is the final velocity, Vi is the initial velocity, g is the acceleration due to gravity, and t is the time.
At the peak, Vf = 0, so we can rearrange the equation to find the time to reach the peak (t_peak):
0 = Vyi - g * t_peak
Solving for t_peak:
t_peak = Vyi / g
t_peak = 17.72 m/s / 9.8 m/s^2
t_peak ≈ 1.81 seconds
2. The maximum height (H) can be calculated using the equation:
H = Vyi * t_peak - (1/2) * g * t_peak^2
H = 17.72 m/s * 1.81 s - (1/2) * 9.8 m/s^2 * (1.81 s)^2
H ≈ 17.97 meters
Therefore, the football reaches a maximum height of approximately 17.97 meters.