Projectile Motion: A Vector Calculus Problem

LearnPlaySolve
8 Sept 202207:32
EducationalLearning
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TLDRThe video script presents a detailed explanation of a projectile motion problem involving a cannon firing a ball towards a target. It uses the principles of gravitational acceleration and trigonometry to derive the optimal angle for firing the cannon. The process involves setting up equations based on the initial velocity and position, and then solving for the angle that allows the projectile to hit a specific target 1000 meters away. The solution involves a quadratic equation in terms of the tangent of the angle and yields two possible angles, with the larger angle being chosen for the demonstration. The video concludes with a call to action for viewers to like and subscribe.

Takeaways
  • 🎯 The problem involves a projectile motion scenario with a cannon firing a ball towards a target 1000 meters away at a height of one meter.
  • 🚀 The cannonball is fired with an initial velocity of 100 meters per second from a height of one meter.
  • 🌐 Gravitational acceleration, approximately 9.8 meters per second squared, acts vertically downwards on the cannonball.
  • 📐 The initial velocity vector is decomposed into horizontal (vx) and vertical (vy) components using trigonometric functions.
  • 🛤️ The position of the cannonball over time is described by parametric equations, forming a parabolic trajectory.
  • 📈 The horizontal position is given by the equation x = 100cos(θ)t, with the initial horizontal position being zero.
  • 📉 The vertical position is given by the equation y = -4.9t^2 + 100sin(θ)t + 1, considering the initial height of one meter.
  • 🎓 The target is located at x = 1000 and y = 1, and the goal is to find the angle θ that results in the parabola intersecting this point.
  • 🔢 By equating the horizontal position equation to 1000 and simplifying, a quadratic equation in terms of the tangent of θ is obtained.
  • 🏆 Two possible angles are found using the quadratic formula, corresponding to two different parabolic trajectories that intersect the target.
  • 🎥 The script concludes with the selection of the larger angle, 50.74 degrees, as the chosen angle to aim the cannon for the best chance of hitting the target.
Q & A

  • What is the average velocity of the cannonball as mentioned in the script?

    -The average velocity of the cannonball is 100 meters per second.

  • What is the height from which the cannonball exits the cannon?

    -The cannonball exits the cannon at a height of one meter.

  • What is the force acting on the cannonball after it is fired?

    -After the cannonball is fired, it is subjected to the force of gravitational acceleration, which is approximately 9.8 meters per second squared acting in the downward vertical direction.

  • What is the initial horizontal velocity (vx) of the cannonball?

    -The initial horizontal velocity (vx) of the cannonball is the constant value derived from the initial velocity vector, which is 100 meters per second multiplied by the cosine of the angle theta (cos θ).

  • What is the initial vertical velocity (vy) of the cannonball?

    -The initial vertical velocity (vy) of the cannonball is the negative value derived from the initial velocity vector, which is 100 meters per second multiplied by the sine of the angle theta (sin θ).

  • How can the position vector be expressed in terms of the initial velocity components?

    -The position vector can be expressed as the antiderivative of the velocity components with respect to time (t). For the horizontal position (x), it is 100cosθ times t plus the initial horizontal position. For the vertical position (y), it is -4.9t squared plus 100sinθ times t plus the initial vertical position.

  • What is the significance of the target being 1000 meters away and one meter high?

    -The target being 1000 meters away and one meter high sets the conditions for the projectile motion problem. It means the cannonball must travel a horizontal distance of 1000 meters and still be at a height of one meter when it reaches the target.

  • How does the angle theta determine the shape of the cannonball's trajectory?

    -The angle theta determines the shape of the cannonball's trajectory by influencing the initial velocity components in both the horizontal and vertical directions. A different angle will result in a different parabolic path, affecting whether the cannonball will hit the target or not.

  • What are the two possible angles that can be used to position the cannon to hit the target?

    -The two possible angles that can be used to position the cannon to hit the target are approximately 39.26 degrees and 50.74 degrees. These angles are derived from solving the quadratic equation in terms of tangent theta.

  • Why are there two possible angles to position the cannon?

    -There are two possible angles to position the cannon because the constraints of the problem allow for two different parabolic trajectories that can intersect the target. Each angle results in a different parabolic path that can potentially hit the target.

  • How can the quadratic formula be used to solve for the tangent of the angle theta?

    -The quadratic formula can be used to solve for the tangent of the angle theta by first expressing the equation in terms of tangent theta. The coefficients a, b, and c from the quadratic equation are used in the formula to find the possible values of tangent theta, which are then used to find the angle theta through the inverse tangent function.

Outlines
00:00
🎯 Projectile Motion and Cannon Targeting

This paragraph introduces a classic physics problem involving projectile motion. It describes a scenario where a cannon is set to fire a cannonball with an initial velocity of 100 meters per second from a height of one meter. The goal is to hit a target 1000 meters away and one meter high. The paragraph delves into the physics of the situation, explaining the role of gravitational acceleration and how to create velocity and position vectors. It uses trigonometric functions to relate the initial velocity components to the angle of the cannon and sets up the problem for solving the optimal firing angle.

05:01
📚 Solving for the Optimal Cannon Angle

The second paragraph continues the projectile motion problem by focusing on the mathematical solution to find the optimal angle for firing the cannon. It explains the process of turning the position vector into a set of parametric equations and how to use these equations to find the angle that will result in a parabola intersecting the target. The paragraph uses the Pythagorean identity and the quadratic formula to solve for the tangent of the angle, presenting two possible solutions. It concludes by selecting the larger angle for the cannon's positioning and encourages viewers to enjoy the process of solving such mathematical problems.

Mindmap
Keywords
💡Projectile Motion
Projectile motion refers to the motion of an object that is launched into the air and is influenced only by gravity. In the context of the video, it is the fundamental concept used to determine the trajectory of the cannonball fired from the cannon. The script discusses how the cannonball's path can be predicted and calculated using the principles of projectile motion, which is essential for hitting the target.
💡Gravitational Acceleration
Gravitational acceleration is the acceleration experienced by objects near the Earth's surface due to the force of gravity. It is approximately 9.8 meters per second squared and acts in the vertical direction, pulling objects downward. In the video, gravitational acceleration is a key factor in determining the y-component of the velocity and position vectors, which affects the trajectory of the cannonball.
💡Velocity Vector
A velocity vector is a vector quantity that describes the rate of change of an object's position, having both magnitude and direction. In the video, the velocity vector is used to represent the initial horizontal and vertical components of the cannonball's velocity, which are essential for calculating its trajectory. The horizontal component (vx) is constant, while the vertical component (vy) changes due to gravity.
💡Anti-Derivative
An anti-derivative, also known as an integral, is a function that represents the area under the curve of another function. In the context of the video, the anti-derivative is used to find the position vector from the velocity vector, which helps in determining the path of the cannonball. The anti-derivative of a constant velocity gives the position, which is essential for creating the parametric equations that describe the projectile's path.
💡Right Triangle
A right triangle is a triangle in which one angle is a right angle (90 degrees). In the video, the concept of a right triangle is used to break down the initial velocity of the cannonball into its horizontal (x) and vertical (y) components. This is done by considering the angle theta, which allows the calculation of the cannon's firing angle for hitting the target.
💡Trigonometric Functions
Trigonometric functions, such as sine, cosine, and tangent, are mathematical functions that relate the angles and sides of a right triangle. In the video, these functions are used to calculate the components of velocity and position vectors, which are crucial for determining the trajectory of the cannonball. The script uses these functions to find the relationship between the angle of the cannon and the horizontal and vertical velocities.
💡Parametric Equations
Parametric equations are equations that describe the relationship between variables using a parameter, usually time (t). In the video, parametric equations are used to express the position of the cannonball at any given time, based on its trajectory. These equations help in visualizing the path of the projectile as a parabola and are essential for solving the problem of finding the correct angle to hit the target.
💡Quadratic Equation
A quadratic equation is a polynomial equation of degree two, typically in the form of ax^2 + bx + c = 0. In the video, the quadratic equation is derived from the parametric equations to solve for the angle theta that will result in the cannonball hitting the target. The quadratic formula is used to find the values of tangent theta, which are then used to determine the angle.
💡Arc Tangent
The arc tangent, or inverse tangent, is the inverse function of the tangent and is used to find the angle whose tangent is a given value. In the video, the arc tangent is used to find the angle theta from the values of tangent theta obtained by solving the quadratic equation. This angle is crucial for positioning the cannon to hit the target.
💡Trajectory
The trajectory is the path followed by an object in motion. In the video, the trajectory of the cannonball is of central importance, as the goal is to find the angle that will result in a trajectory that intersects the target. The script discusses how the trajectory, represented by a parabola, is influenced by the angle at which the cannon is fired.
💡Optimization
Optimization involves finding the best solution or solution set from a set of possible solutions, often by maximizing or minimizing a particular value. In the context of the video, optimization is the process of determining the optimal angle for the cannon to hit the target at a distance of 1000 meters. The script uses mathematical analysis and equations to optimize the cannon's firing angle.
Highlights

Cannonball is fired at an average velocity of 100 meters per second from a height of one meter.

Projectile motion problem is solved using gravitational acceleration, which is approximately 9.8 meters per second squared.

The x-component of acceleration is zero, and the y-component is -9.8 due to Earth's gravity.

Initial horizontal velocity is represented by a constant, and the initial vertical velocity is the anti-derivative of -9.8.

The cannonball's position vector is derived from the antiderivative of its velocity vector.

The parabola's shape is determined by the angle theta at which the cannon is positioned.

The target is 1000 meters away and one meter high, which defines the desired impact point.

The initial horizontal position is set to zero for simplicity.

The initial vertical position is one meter, as the cannon fires at a height of one meter.

The position vector is turned into a set of parametric equations with time t as the parameter.

The equation of motion is simplified and rewritten in terms of a single trigonometric function.

A quadratic equation in terms of tangent theta is formed to find the optimal angle for firing.

The quadratic formula is used to solve for tangent theta, yielding two possible angles for theta.

The two angles, approximately 39.26 degrees and 50.74 degrees, represent two different parabolic trajectories that could hit the target.

The larger angle of 50.74 degrees is chosen for positioning the cannon to test the calculations.

The process demonstrates the practical application of physics in solving real-world problems like hitting a target with a cannon.

Transcripts

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Physics ProblemTrigonometryProjectile MotionCannon TargetingOptimal AngleGravitational AccelerationMathematical SolutionEducational ContentScience DemonstrationPhysics Calculation