Projectile Motion: Shoot the Monkey
TLDRIn this physics-based problem, the video explores the optimal angle and initial velocity required to hit a monkey hanging from a branch with a Nerf bullet. The monkey is at a height H above the ground and the hunter is at a distance D away. By setting up equations of motion for both the monkey and the bullet and solving for the angle, it's determined that the tangent of the angle is equal to H/D. The video also addresses different scenarios, such as hitting the monkey halfway to the ground and just before it lands, with varying initial velocities required for each case. The problem emphasizes the importance of understanding projectile motion and the application of trigonometry and physics principles.
Takeaways
- π― The problem involves calculating the angle and initial velocity required to hit a monkey hanging from a branch with a projectile.
- πΉ The monkey is in free fall once the gun is fired, and the bullet follows projectile motion.
- π The position of the monkey as a function of time is given by y = H - (9.8 * t^2), considering the ground as the reference point.
- π The bullet's motion is described by two equations: one for the x-coordinate (X = v0x * t) and one for the y-coordinate (Y = v0y * t - (9.8 * t^2)).
- π The initial velocity of the bullet is broken down into x and y components using trigonometry: v0x = v0 * cos(ΞΈ) and v0y = v0 * sin(ΞΈ).
- π To hit the monkey, the x and y coordinates of the bullet must align with those of the monkey at the same time.
- π The angle ΞΈ can be found using the tangent function: tan(ΞΈ) = H/D, where H is the height and D is the distance from the monkey.
- π’ For the first scenario, the bullet must be fired at an angle of 26.6 degrees to hit the monkey as it falls from a height of 2 meters with a distance of 4 meters.
- π¦ The initial velocity required to hit the monkey halfway between its starting point and the ground is approximately 9.9 m/s.
- π To hit the monkey just before it hits the ground, the minimum initial velocity required is 7 m/s.
Q & A
What is the main problem presented in the video?
-The main problem is to determine the angle and initial velocity required to hit a monkey hanging from a branch with a Nerf bullet fired from a distance.
What is the initial position of the monkey in the coordinate system used?
-The initial position of the monkey is at height H above the ground, with the ground set to zero on the vertical axis.
What is the acceleration due to gravity used in the calculations?
-The acceleration due to gravity used is 9.8 m/s^2, denoted as little G.
How is the initial velocity of the Nerf bullet broken down for the equations of motion?
-The initial velocity of the Nerf bullet is broken down into an X component (v0 cosine of Theta) and a Y component (v0 sine of Theta) using trigonometry.
What is the relationship between the angle of firing, the height, and the distance in the problem?
-The tangent of the angle of firing (Theta) is equal to the height (H) divided by the distance (D). This relationship helps in determining the angle for a direct hit.
What is the initial velocity required to hit the monkey halfway between its starting point and the ground?
-The initial velocity required is approximately 9.9 m/s, calculated by considering the height H as 2 m and the distance D as 4 m.
What is the minimum initial velocity needed to hit the monkey just before it hits the ground?
-The minimum initial velocity needed is 7 m/s, which is derived from the equations of motion and the given parameters.
How does the initial velocity affect the time it takes to hit the monkey?
-As the initial velocity increases, the time it takes for the bullet to hit the monkey decreases, approaching an instant hit in the limit of very high velocities.
What is the significance of the line of sight in this problem?
-The line of sight is crucial as it determines the angle at which the bullet must be fired. As long as the shooter aims along the line of sight, a hit is guaranteed if the bullet has enough range to reach the monkey.
How does the height of the monkey affect the required initial velocity?
-The height of the monkey affects the time the monkey takes to fall, which in turn influences the required initial velocity to meet the monkey at a specific point in its descent. Higher initial velocities are needed if the monkey is higher up.
What is the role of projectile motion in solving this problem?
-Projectile motion principles are essential in determining the path of the Nerf bullet and how it will intersect with the monkey's falling path. The equations of motion for projectile motion are used to calculate the required angle and velocity.
Outlines
π― Introduction to the Physics Problem: Shooting the Monkey
The video begins by introducing the audience to a physics problem involving a monkey hanging from a branch and a hunter attempting to shoot the monkey from a distance. The main challenge is to determine the correct angle and initial velocity for the bullet to hit the monkey. The problem is set up with the monkey at a height H above the ground and the hunter at a distance D away. The video aims to solve three questions: the angle required to hit the monkey, the initial velocity needed to hit it halfway between its starting point and the ground, and the initial velocity to hit it just before it lands.
π Setting up the Equations of Motion
The video proceeds to set up the equations of motion for both the monkey and the bullet. The monkey's motion is described by a simple freefall equation, with its position given as a function of time. The bullet's motion, however, requires equations in both the X and Y directions due to its projectile motion. The initial velocity of the bullet is decomposed into horizontal (X-direction) and vertical (Y-direction) components using trigonometry. The equations are then used to describe the position of both objects as a function of time.
π’ Solving for the Angle and Initial Velocity
The video explains the process of solving for the angle and initial velocity required to hit the monkey. By equating the positions of the monkey and the bullet at the same time, the video derives an equation for the angle in terms of the height (H) and distance (D). The tangent of the angle is found to be equal to H/D. The video then goes on to solve for the initial velocity required to hit the monkey at different heights, using algebraic manipulation and substitution of known values.
π Demonstrating the Projectile Motion with Simulation
The video concludes by demonstrating the projectile motion using a simulation. The bullet and the monkey's positions are shown over time, with the bullet following a parabolic trajectory and the monkey in freefall. The simulation visually confirms the calculations made earlier in the video, showing the bullet hitting the monkey at the calculated angles and velocities. The video emphasizes the importance of understanding the physics concepts involved and the practical application of these concepts in solving real-world problems.
Mindmap
Keywords
π‘Projectile Motion
π‘Freefall
π‘Acceleration due to Gravity
π‘Trigonometry
π‘Angle of Launch
π‘Initial Velocity
π‘Trajectory
π‘Equations of Motion
π‘Tangent
π‘Simulation
π‘Line of Sight
Highlights
The problem involves a monkey hanging on a branch and a hunter aiming to shoot the monkey with a Nerf bullet.
The monkey is above the ground at a height H, and the hunter is at a distance D away from the monkey.
The key to solving the problem is to determine the correct angle and initial velocity to fire the Nerf bullet so it hits the monkey.
Upon firing the gun, the monkey will let go of the branch in an attempt to avoid being shot, initiating freefall motion.
The equation of motion for the monkey is based on freefall, with an initial position of H and an acceleration due to gravity of 9.8 m/s^2.
The bullet's motion is a combination of horizontal and vertical projectile motion, with its initial velocity broken into x and y components.
The x-component of the bullet's initial velocity is given by v0 * cos(ΞΈ), and the y-component by v0 * sin(ΞΈ), where ΞΈ is the angle to be determined.
To hit the monkey, the bullet and the monkey must coincide in both x and y positions, leading to a system of equations to solve for ΞΈ and the initial velocity.
The tangent of the angle ΞΈ is found to be equal to H/D, which simplifies the process of finding the angle for the optimal shot.
The initial velocity required to hit the monkey at a height of H/2 is derived and calculated based on the given values of H and D.
For the bullet to hit the monkey just before it hits the ground, a minimum initial velocity is calculated, ensuring the bullet has enough range to reach the monkey.
The problem demonstrates the application of projectile motion and freefall in a real-world scenario, showcasing the importance of understanding these physical concepts.
The video includes a simulation to visually demonstrate the projectile motion and the timing required to hit the monkey at different heights.
The final velocity required to hit the monkey at the very moment it touches the ground is found to be at least 7 m/s.
In the limit of very high initial velocities, the bullet effectively follows the line of sight, hitting the monkey almost instantaneously.
The video concludes with a practical demonstration of the theoretical concepts, reinforcing the learning experience through visual aid.
Transcripts
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