Physics 3.5.4b - Projectile Practice Problem 2
TLDRThe video script presents a physics problem involving Farmer Bob and his horse Flash jumping across a 100-meter wide canyon in 5 seconds. The challenge is to calculate the initial velocity and the angle of the jump. By analyzing the horizontal and vertical components of the velocity independently, the script demonstrates how to use equations of motion to find the horizontal component (20 m/s) and the vertical component (24.5 m/s). Combining these with the Pythagorean theorem, the script concludes that the initial velocity is 31.6 m/s at an angle of 50.8 degrees above the horizontal, providing a clear and engaging explanation of projectile motion.
Takeaways
- π The problem involves calculating the initial velocity of Farmer Bob and his horse Flash as they jump across a 100-meter wide canyon in 5 seconds.
- π The initial velocity has both horizontal (v0x) and vertical (v0y) components, which are essential for a successful projectile motion.
- π The horizontal component of the initial velocity (v0x) is determined to be 20 m/s using the distance traveled and the time in air.
- π For the vertical component (v0y), the problem considers the time from the start to the peak of the jump, which is half the total time of 5 seconds, i.e., 2.5 seconds.
- π The vertical motion is analyzed with the final velocity at the peak being zero (no upward or downward motion at the highest point).
- π‘ The vertical component of the initial velocity (v0y) is calculated to be 24.5 m/s using the equation v = v0 + at, with a being the acceleration due to gravity (-9.8 m/s^2) and t being the time to the peak (2.5s).
- π’ The magnitude of the initial velocity (v0) is found using the Pythagorean theorem, combining the horizontal and vertical components, resulting in 31.6 m/s.
- π The angle (ΞΈ) of the initial velocity vector above the horizontal is determined using the tangent function, with ΞΈ being 50.8 degrees.
- π― The problem-solving approach requires understanding the independence of horizontal and vertical motions in projectile problems.
- π The key to solving the problem is accurately calculating both the horizontal and vertical components of the initial velocity and then combining them to find the magnitude and direction.
- π The script provides a step-by-step guide to solving physics problems involving projectile motion, emphasizing the importance of breaking down the motion into its components.
Q & A
What is the total distance that Farmer Bob and his horse Flash need to jump across the canyon?
-The total distance that Farmer Bob and his horse Flash need to jump across the canyon is 100 meters.
How long does it take for Farmer Bob and his horse to cross the canyon?
-It takes 5 seconds for Farmer Bob and his horse to cross the canyon.
What is the horizontal component of the initial velocity (v0x) calculated to be?
-The horizontal component of the initial velocity (v0x) is calculated to be 20 m/s.
How is the vertical component of the initial velocity (v0y) determined?
-The vertical component of the initial velocity (v0y) is determined by considering the motion from the start to the peak of the jump, where the final velocity is zero, and using the equation v = v0 + at, where a is the acceleration due to gravity (-9.8 m/s^2) and t is the time (2.5 seconds).
What is the magnitude of the vertical component of the initial velocity (v0y)?
-The magnitude of the vertical component of the initial velocity (v0y) is 24.5 m/s.
How can the magnitude of the total initial velocity (v0) be calculated?
-The magnitude of the total initial velocity (v0) can be calculated using the Pythagorean theorem with the horizontal and vertical components (v0x and v0y). v0 = sqrt((v0x)^2 + (v0y)^2), which results in a value of 31.6 m/s.
What is the angle (theta) at which Farmer Bob and his horse jump?
-The angle (theta) at which Farmer Bob and his horse jump is 50.8 degrees above the horizontal, calculated using the tangent function and the ratio of the vertical component to the horizontal component of the initial velocity.
Why is it necessary to consider both horizontal and vertical components of the initial velocity in this problem?
-It is necessary to consider both horizontal and vertical components of the initial velocity because the problem involves projectile motion, where the motion can be analyzed separately along horizontal and vertical axes. This allows us to find the individual components and then combine them to determine the total initial velocity vector.
What is the significance of the time it takes to reach the peak of the jump?
-The time it takes to reach the peak of the jump is significant because at the peak, the vertical velocity is zero, allowing us to analyze the motion from the start to the peak independently. This helps in determining the vertical component of the initial velocity.
How does the acceleration due to gravity affect the calculation of the initial velocity components?
-The acceleration due to gravity, which is -9.8 m/s^2, affects the calculation of the initial velocity components by being the acceleration term in the kinematic equations used to determine the vertical component of the initial velocity. It is a crucial factor in the analysis of projectile motion.
What is the role of the Pythagorean theorem in solving this problem?
-The Pythagorean theorem is used to find the magnitude of the total initial velocity by calculating the hypotenuse of the right triangle formed by the horizontal and vertical components of the initial velocity. It allows us to combine these components to find the overall initial velocity vector.
Why is the tangent function used to find the angle of the initial velocity vector?
-The tangent function is used to find the angle of the initial velocity vector because it relates the ratio of the opposite side (v0y) to the adjacent side (v0x) in a right triangle. This ratio is directly related to the angle (theta) at which the initial velocity vector is inclined above the horizontal.
Outlines
π Initial Conditions and Horizontal Motion Analysis
This paragraph introduces the problem of Farmer Bob and his horse Flash jumping across a 100-meter wide canyon in 5 seconds. The goal is to determine the initial velocity's magnitude and direction, specifically the angle (Theta) of inclination above the horizontal. The initial velocity vector (v0) is broken down into its horizontal (v0x) and vertical (v0y) components. The horizontal motion is first analyzed, considering no horizontal acceleration and using the equation of motion to find the horizontal velocity (v0x = 20 m/s). This value is identified as the horizontal component of the initial velocity.
π Vertical Motion Analysis and Initial Velocity Calculation
The second paragraph delves into the vertical motion analysis to find the vertical component of the initial velocity (v0y). The vertical motion is considered positive, with acceleration due to gravity (9.8 m/s^2) acting in the opposite direction. The time to reach the peak of the jump is calculated to be 2.5 seconds. Using the final vertical velocity at the peak (zero), the vertical component of the initial velocity (v0y = 24.5 m/s) is determined. Finally, the magnitude of the initial velocity (v0 = 31.6 m/s) and the angle (Theta = 50.8 degrees above the horizontal) are calculated using the Pythagorean theorem and the tangent function, respectively.
Mindmap
Keywords
π‘Projectile Motion
π‘Initial Velocity
π‘Horizontal Component
π‘Vertical Component
π‘Acceleration
π‘Time of Flight
π‘Parabolic Path
π‘Velocity Vector
π‘Angle (Theta)
π‘Pythagorean Theorem
π‘Tangent Function
Highlights
Problem setup: Farmer Bob and his horse Flash jump a 100-meter wide canyon in 5 seconds.
Objective: Find the initial velocity's magnitude and direction (angle Theta) for the jump.
The jump is modeled as a projectile motion with an initial velocity vector.
Horizontal component (v0x) of the initial velocity is calculated first.
Since there's no horizontal acceleration, the horizontal motion simplifies to distance (X) equals v0x times time (T).
The horizontal component of the initial velocity (v0x) is found to be 20 m/s.
Both horizontal and vertical components of the initial velocity must be calculated for the full vector.
Vertical motion analysis begins with considering the time to reach the peak of the jump, which is 2.5 seconds.
At the peak, the vertical velocity is zero, and the horse is moving horizontally.
The vertical component (v0y) of the initial velocity is found using the equation v = v0 - gt, resulting in 24.5 m/s.
Combining the horizontal and vertical components using the Pythagorean theorem gives the magnitude of the initial velocity.
The magnitude of the initial velocity (v0) is calculated to be 31.6 m/s.
The angle Theta is determined using the tangent function, with Theta being 50.8 degrees above the horizontal.
The problem demonstrates the application of projectile motion equations in real-world scenarios.
The solution process highlights the importance of analyzing horizontal and vertical components independently.
The method used can be applied to various physics problems involving motion and vectors.
The problem showcases the practical use of the Pythagorean theorem in calculating vector magnitudes.
Understanding the relationship between initial velocity components and the resultant vector is crucial.
The solution involves a step-by-step approach, making it easier to follow and comprehend.
The problem emphasizes the significance of time in analyzing projectile motion.
The angle of inclination is a key factor in determining the trajectory of the projectile.
The problem illustrates the application of basic physics principles to more complex scenarios.
Transcripts
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