A Range Equation Problem with Two Parts

Flipping Physics
19 Jun 201407:22
EducationalLearning
32 Likes 10 Comments

TLDRIn the video, Mr. P attempts to throw a ball into a bucket. Initially, the ball falls short by 34 cm. Using the Range Equation, the team calculates the required initial speed and launch angle for a successful throw. They find that the second attempt should be thrown at 43 degrees, but also note that complementary angles (47 degrees) would work due to the symmetry of projectile motion. The video combines physics problem-solving with a playful tone, culminating in a practical demonstration of the solution.

Takeaways
  • 🎾 Mr. P attempts to throw a ball into a bucket 581 centimeters away with an initial angle of 55 degrees, but falls short by 34 centimeters.
  • 📐 The Range Equation is used to solve projectile motion problems where the displacement in the Y direction is zero, making it suitable for this scenario.
  • 🔢 The first range (first attempt) is calculated by subtracting the short distance (34 cm) from the total distance (581 cm), converting it to meters (5.47 m).
  • 🚀 The magnitude of the initial velocity (v_sub_i) is determined by using the Range Equation and known values, resulting in 7.55675 meters per second.
  • 🔄 For the second attempt, the initial speed remains the same, and the total distance is the bucket's distance (5.81 meters).
  • 📐 The Range Equation is again utilized, this time solving for the launch angle of the second attempt, which is found to be approximately 43 degrees.
  • 🏀 Complementary angles (angles that add up to 90 degrees) are acknowledged as both providing the same range, suggesting an alternative launch angle of 47 degrees.
  • 🤓 The problem-solving process emphasizes the importance of understanding the physics behind projectile motion and the application of the Range Equation.
  • 👥 Collaboration and dialogue between characters help to clarify concepts and steps in solving the problem, highlighting the value of teamwork in problem-solving.
  • 🎥 The video script format provides a narrative structure to learning, making complex concepts more accessible and engaging for the audience.
  • 🌟 The video concludes with a practical demonstration of the solution, reinforcing the theoretical concepts with a real-world application.
Q & A

  • What is the problem that Mr. P is trying to solve?

    -Mr. P is trying to find the correct angle to throw a ball so that it lands in a bucket located 581 centimeters away from him horizontally, after having failed to do so with an initial throw at a 55-degree angle.

  • What is the significance of the bucket's height in this problem?

    -The height of the bucket is significant because the ball is always released at the same height as the top of the bucket, which means the displacement in the Y direction is zero, allowing the use of the Range Equation.

  • What is the Range Equation mentioned in the script?

    -The Range Equation states that the range of a projectile equals the magnitude of the initial velocity (or initial speed) squared, times the sine of two times the launch angle, divided by the acceleration due to gravity.

  • How do they determine the initial speed for the first throw?

    -They use the Range Equation for the first throw, knowing the first range (547 cm), the first launch angle (55 degrees), and the acceleration due to gravity (9.81 m/s^2), and solve for the unknown initial speed (v_sub_i).

  • What is the calculated initial speed for Mr. P's throws?

    -The calculated initial speed is approximately 7.55675 meters per second, found by taking the square root of the product of the acceleration due to gravity and the first range, divided by the sine of two times the first launch angle.

  • What is the total distance to the bucket for the second attempt?

    -The total distance to the bucket for the second attempt is 581 centimeters, which is converted to 5.81 meters for use in the Range Equation.

  • How is the launch angle for the second attempt determined?

    -The launch angle for the second attempt is determined by rearranging the Range Equation to solve for the sine of two times the launch angle, and then taking the inverse sine. The calculated angle is 43 degrees, but it is noted that the complementary angle of 47 degrees would also work due to the property of complementary angles having the same range.

  • What is the diameter of the basketball used in the problem?

    -The diameter of the basketball used in the problem is 23 and a half centimeters.

  • What is the diameter of the bucket in the problem?

    -The diameter of the bucket is 30 centimeters.

  • Why are there two correct launch angles for this problem?

    -There are two correct launch angles because complementary angles, which add up to 90 degrees, will both result in the same range when thrown.

  • What is the final answer for the launch angle that will land the ball in the bucket?

    -The final answer for the launch angle is either 43 or 47 degrees, as both angles are complementary and will result in the ball landing in the bucket.

Outlines
00:00
📚 Solving the Range Equation Problem

This paragraph introduces a two-step Range Equation problem where Mr. P attempts to throw a ball into a bucket. The first throw falls short, and the group discusses using the Range Equation to determine the correct angle for the second throw. The problem is set up with the given conditions: initial horizontal distance, angle of projection, and the fact that the ball is released at the same height as the top of the bucket. The Range Equation is derived and used to calculate the unknown initial speed, which remains constant for both throws. The process of solving for the launch angle is explained, with the group working through the physics and math involved.

05:01
🎯 Calculating the Correct Launch Angle

In this paragraph, the group continues their discussion on the Range Equation, focusing on solving for the launch angle that will allow the ball to land in the bucket. They use the previously calculated initial speed and apply it to the second attempt, where the range is known to be the total distance to the bucket. The calculations lead to a specific angle, but the group also considers the possibility of complementary angles providing the same range, leading to two potential correct answers. The paragraph concludes with a practical demonstration of the solution and a reflection on the learning process.

Mindmap
Keywords
💡Range Equation
The Range Equation is a fundamental concept in physics used to calculate the horizontal distance (range) that a projectile will travel when thrown at a certain angle and initial speed. In the video, it is used to determine the launch angle needed for Mr. P to successfully throw a ball into a bucket. The equation is given by the formula: Range = (initial velocity)^2 * sin(2 * angle) / g, where g is the acceleration due to gravity. The video demonstrates the application of this equation in solving real-world problems, specifically in the context of projectile motion.
💡Projectile Motion
Projectile Motion refers to the motion of an object that is launched into the air and moves under the influence of gravity and air resistance, if considered. In the video, Mr. P's ball throwing attempt is an example of projectile motion, as the ball is thrown with an initial velocity at an angle to the horizontal and follows a parabolic trajectory. The principles of projectile motion are essential in understanding how to adjust the throw to land the ball in the bucket.
💡Initial Velocity
Initial velocity is the speed at which an object begins its motion. In the context of the video, it is the speed with which Mr. P throws the ball. The magnitude of the initial velocity is a crucial factor in the Range Equation, as it directly affects the distance the ball will travel. The video emphasizes that the initial speed remains constant in both attempts to throw the ball.
💡Launch Angle
The launch angle is the angle between the initial velocity vector and the horizontal direction. In the video, finding the correct launch angle is key to ensuring the ball lands in the bucket. The launch angle is adjusted based on the results from the Range Equation and the known initial velocity to achieve the desired outcome.
💡Acceleration Due to Gravity
Acceleration due to gravity, often denoted as 'g', is the acceleration that an object experiences due to the gravitational pull of the Earth. In the context of the video, it is a constant value used in the Range Equation to calculate the projectile's motion. The acceleration due to gravity is approximately 9.81 meters per second squared on the surface of the Earth.
💡Displacement
Displacement refers to the change in position of an object. In the video, displacement is used to describe the horizontal distance (range) the ball travels from the point of release to the bucket. The concept of displacement is essential in the Range Equation as it helps in calculating the trajectory of the projectile.
💡Trigonometry
Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. In the video, trigonometry is used to calculate the sine of the launch angle, which is a component of the Range Equation. The application of trigonometric functions is crucial in understanding and solving problems related to projectile motion.
💡Complementary Angles
Complementary angles are two angles whose sum is 90 degrees. In the context of the video, it is mentioned that complementary angles can yield the same range in projectile motion. This concept is important as it provides an alternative solution for the launch angle, indicating that both angles adding up to 90 degrees will result in the ball landing at the same distance.
💡Inverse Sine
The inverse sine function, often denoted as sin^(-1) or arcsin, is a mathematical operation that returns the angle whose sine is a given value. In the video, the inverse sine is used to solve for the launch angle from the Range Equation. It is a crucial step in determining the precise angle needed for Mr. P to successfully throw the ball into the bucket.
💡Parabolic Trajectory
A parabolic trajectory is the path followed by an object in free-fall, where the only force acting on it is gravity. In the video, the ball's flight path after being thrown is a parabolic shape due to the influence of gravity. Understanding the parabolic trajectory is essential for analyzing projectile motion and predicting where the ball will land.
💡Projectile Range
Projectile range is the horizontal distance that a projectile travels from the point of release to the point where it lands. In the video, the range is a critical concept as the goal is to make the ball land within a specific distance, i.e., in the bucket. The range is directly calculated using the Range Equation, which takes into account the initial velocity, launch angle, and acceleration due to gravity.
Highlights

Mr. P throws a ball toward a bucket, which is 581 centimeters away.

The ball is thrown at an initial angle of 55 degrees above the horizontal.

The ball falls 34 centimeters short of the bucket on the first throw.

The problem requires finding the angle to throw the ball so it lands in the bucket when the initial speed is the same.

The Range Equation is identified as the appropriate method to solve the problem.

The displacement in the Y direction is zero for the second throw, allowing the use of the Range Equation.

The range for the first throw is calculated as 547 centimeters.

The first throw's range is converted to meters for calculation purposes.

The magnitude of the initial velocity is the unknown variable for the first throw.

The initial speed is determined to be 7.55675 meters per second for the first throw.

The second throw's range is the total distance to the bucket, 5.81 meters.

The launch angle for the second throw is calculated using the Range Equation.

The calculated launch angle for the second throw is 43 degrees.

It is noted that complementary angles can give the same range, suggesting two possible correct angles.

The correct answer includes either 43 or 47 degrees as the launch angle for the second throw.

The practical application of the problem involves throwing a basketball into a bucket.

The problem-solving process is demonstrated through a collaborative and engaging conversation.

Transcripts

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