A Football Team Is In A Position To Kick A Field Goal To Win

1a Football Team Is In A Position To Kick A Field Goal To Win A Game

A football team is in a position to kick a field goal to win a game. The ball is placed 36 m (approximately 39 yards) from the goalposts. The kicker kicks the ball with a resultant velocity of 20 m/s at an angle of 33°. List your knowns: (2 pts) What was the initial horizontal velocity? (2 pts) What was the initial vertical velocity? (2 pts) How long was the ball in the air ( total flight time)? (4 pts) What was the horizontal distance traveled? Did they win? (4 pts) Calculate the following quantities for the diagram shown below: (16 points total) a. The angular velocity at the hip over each time interval (6 points) b. The angular velocity at the knee over each time interval (6 points) Would it be meaningful information to calculate the average angular velocities at the hip and knee for the movement shown? Provide a rationale for your answer. (4 points) A soccer ball is kicked from the playing field. If the ball is in the air for 2.2 s (total flight time), what is the maximum height achieved? (4 points) (neglect air resistance) Two cyclists B 1 and B 2 ) are racing at exactly the same velocity (say, 12 m/s) and come to a curve in the road (point A). At this point they are tied. Throughout the first half of the curve (points A-C), it appears that the cyclist in the outside lane ( B2 ) remains tied with the cyclist in the inside lane ( B1 ). Assume that the cyclist in the inside lane (B1) maintains a constant velocity. Using terms such as “constantâ€, “zeroâ€, “sameâ€, “increaseâ€, “decreaseâ€, “positiveâ€, “negative†etc. answer the following questions: a) What are the differences (if any) between the linear distances traveled by the cyclists between points A and C. List the equation that explains this. (4 points) B1 B1 B2 B1 B2 B2 b) What are the differences (if any) between the tangential (linear) velocities of the cyclists at points A and C. List the equation that explains this. (4 points) c) What are the differences (if any) between the tangential (linear) accelerations of the cyclists between points A and C. List the equation that explains this. (4 points) d) What is the difference (if any) between the radial acceleration of B1 at points A and C. What is the difference (if any) between the radial acceleration of B2 at points A and C. List the equation that explains this. (4 points)

Sample Paper For Above instruction

The problem of a football kick involves analyzing projectile motion to determine the initial velocities, flight time, and distance traveled, which are crucial in understanding whether the kick successfully reaches the goalposts. The knowns include the initial resultant velocity of 20 m/s and the angle of projection at 33°. The problem requires decomposing this velocity into horizontal and vertical components, calculating the total time the ball remains airborne, and estimating the horizontal distance covered.

Starting with the known initial velocity (v = 20 m/s) and the angle of projection (θ = 33°), the initial horizontal velocity (v_x) can be determined using the cosine component:

v_x = v cos(θ) = 20 cos(33°) ≈ 20 * 0.839 = 16.78 m/s

Similarly, the initial vertical velocity (v_y) is given by the sine component:

v_y = v sin(θ) = 20 sin(33°) ≈ 20 * 0.545 = 10.9 m/s

To find the total flight time (t), consider the vertical motion under gravity (g ≈ 9.81 m/s²). The time to reach the maximum height (t_up) is:

t_up = v_y / g ≈ 10.9 / 9.81 ≈ 1.11 seconds

Since the flight is symmetrical, total time:

t_total = 2 * t_up ≈ 2.22 seconds

Given the horizontal velocity remains constant (neglecting air resistance), the horizontal distance (range) is:

Range = v_x t_total ≈ 16.78 2.22 ≈ 37.2 meters

Because the distance from the ball's position to the goalposts is 36 meters, the ball would just cross the goal line, implying a successful field goal attempt.

Additional Kinematic and Biomechanical Analyses

The questions regarding angular velocities at the hip and knee, and the maximum height of a soccer ball involve separate physics principles such as rotational kinematics and projectile motion. For angular velocities, the variations over time depend on joint acceleration patterns, which are typically measured during movement analyses. The maximum height of the soccer ball is found through vertical displacement calculations similar to the football's vertical component, yielding a maximum height of:

h_max = (v_y)^2 / (2 g) ≈ (10.9)^2 / (2 9.81) ≈ 6.06 meters

Racing Cyclists Analysis

The comparison between cyclists in lanes B1 and B2 involves analyzing their linear and tangential velocities and accelerations while navigating a curved path. Since B1 maintains a constant velocity, its tangential velocity remains constant, whereas B2's motion may involve changes in speed depending on the curvature and forces involved. The linear distance traveled is given by the product of velocity and time, with the equation:

s = v * t

which indicates that if both start from the same point and maintain constant speed, their traveled distances depend linearly on time. The tangential and radial acceleration analyses require calculus to evaluate the changes in velocity and direction, reflecting the forces experienced during the curved motion.

Conclusion

Analyzing projectile motion and rotational kinematics in sport scenarios provides critical insights into movement efficiency and success probability. The calculations demonstrate fundamental physics principles applied in sports science and biomechanics, essential for coaching, injury prevention, and performance optimization.

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