A guy playing basket-ball needs to make the penalty shot for his team to win. Given that the ball is thrown from a height of 2 meters at an angle of 60 degrees, that the post is situated at a distance of 7 m and standing at 3.5 m:
a) What does the initial velocity have to be?
b) What is the balls apogee?
c) What is the flight time to the basket?
I'm not sure what formulas to choose, etc. The method is not clear to me.
To solve these questions, we can use the principles of projectile motion. Projectile motion describes the motion of an object that is influenced by both vertical and horizontal forces. Here's how to approach each question:
a) To find the initial velocity, we can divide the motion into horizontal and vertical components. The horizontal component of the velocity remains constant throughout the motion, while the vertical component is influenced by gravity.
1. First, calculate the vertical component of the initial velocity using the given angle. The vertical component (V_y) can be found using the formula V_y = V * sin(θ), where V is the initial velocity and θ is the angle (60 degrees in this case).
2. Next, find the time it takes for the ball to reach its maximum height. The time (t) can be found using the formula t = V_y / g, where g is the acceleration due to gravity (approximately 9.8 m/s^2).
3. Now, calculate the horizontal component of the initial velocity (V_x). The horizontal distance (d) covered is given as 7 m, and the time taken (t) is already known. So, V_x = d / t.
4. Finally, find the magnitude of the initial velocity by using the Pythagorean theorem: V = √(V_x^2 + V_y^2).
b) The apogee is the highest point in the projectile's trajectory, where its vertical velocity becomes zero. At the apogee, the gravitational force pulls the projectile downward, gradually decreasing its vertical velocity until it reaches zero.
1. Use the formula for vertical displacement (h) to find the maximum height. The formula is h = V_y^2 / (2g).
c) The flight time is the total time the ball is in the air. It can be calculated by considering the time it takes to reach the maximum height and the time it takes to descend from the maximum height to the basket.
1. Multiply the time to reach the maximum height (t) by 2 to account for the descent. The total flight time is 2t.
By following these steps and plugging in the given values (height, angle, distance, standing height, and gravitational acceleration), you should be able to find the answers to each question.