trying to win the superbowl, a placekicker attempts a field goal from 45 m away. when kicked, the ball leaves the ground with a speed of 22m/s at an angle of 50 degrees above the horizontal. what can you say about the ball when it reaches the crossbar that is 3.0m above the ground?
The vertical velocity component starts out at 22*sin50 = 16.85 m/s
The horizontal velocity component remains 22*cos50 = 14.14
The time to reach the goal post is
T = 45/14.14 = 3.18 s
The height of the ball at that time is
Y = 16.85 T - (g/2)*T^2
= 53.6 - 49.6 = 4.0 m
Field Goal!
so does the ball clear by 1 m or does the ball not clear the crossbar and misses it by 1 m.
To determine what happens to the ball when it reaches the crossbar, we need to analyze its motion in terms of horizontal and vertical components. Let's break down the problem step by step:
1. Decompose the initial velocity: The ball is kicked with an initial speed of 22 m/s at an angle of 50 degrees above the horizontal. We can break this velocity into horizontal and vertical components:
- The horizontal component (Vx) can be found using the equation: Vx = V * cos(theta), where V is the initial speed and theta is the angle.
- The vertical component (Vy) can be found using the equation: Vy = V * sin(theta).
Plugging in the values, we get:
Vx = 22 m/s * cos(50 degrees) ≈ 14.17 m/s
Vy = 22 m/s * sin(50 degrees) ≈ 16.87 m/s
2. Determine the time of flight: The time it takes for the ball to reach the crossbar can be found using the vertical component of velocity. We can use the equation:
- d = Vyt - (1/2)gt^2, where d is the vertical displacement (3.0 m above the ground), Vy is the initial vertical velocity, g is the acceleration due to gravity (≈ 9.8 m/s^2), and t is the time of flight.
Rearranging the equation, we get:
(1/2)gt^2 - Vyt + d = 0
Plugging in the values, the equation becomes:
(1/2)(9.8 m/s^2)t^2 - (16.87 m/s)t + 3.0 m = 0
We can solve this quadratic equation using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values, we get:
t = (-(-16.87) ± √((-16.87)^2 - 4 * 0.5 * 9.8 * 3.0)) / (2 * 0.5)
Solving the equation, we find two possible values for t: t ≈ 1.16 s and t ≈ 2.23 s.
3. Analyze the horizontal displacement: Once we have the time of flight, we can determine the horizontal displacement of the ball.
The horizontal displacement (dx) can be found using the equation:
dx = Vx * t
Plugging in the values, we get:
dx = 14.17 m/s * t
For t ≈ 1.16 s, dx ≈ 14.17 m/s * 1.16 s ≈ 16.43 m
For t ≈ 2.23 s, dx ≈ 14.17 m/s * 2.23 s ≈ 31.64 m
4. Evaluate the result: Comparing the horizontal displacement to the field goal distance of 45 m, we can say:
- For t ≈ 1.16 s, the ball falls short of the crossbar as the horizontal displacement (16.43 m) is less than the field goal distance (45 m).
- For t ≈ 2.23 s, the ball clears the crossbar as the horizontal displacement (31.64 m) is greater than the field goal distance (45 m).
Therefore, when the ball reaches the crossbar that is 3.0 m above the ground, it will clear the crossbar only if the time of flight is approximately 2.23 seconds or longer. Otherwise, it will fall short.