Exploring Real-Life Projectile Motion: Peyton Manning's Epic Throw
Written on
Chapter 1: Introduction to the Throw
When you have ample resources and a desire for excitement, what do you do? One possibility is to recreate a vintage football stunt, this time featuring the talents of Peyton Manning and Chris Carter. The premise? Manning launched a football from a towering 23-story building to a receiver waiting in a nearby park. Check out the full video for the thrilling visuals.
This entertaining clip isn't just about the fun; it poses an intriguing physics challenge with multiple layers to explore. Let's delve into the fundamental aspects, beginning with the assumption of no air resistance. Even under this simplified condition, numerous questions arise:
- How high did Peyton actually throw the ball? This detail isn't specified.
- Is it feasible for an average person to achieve a throw that covers a horizontal distance of 85 yards, as claimed?
- Is a launch angle of 45 degrees the optimal choice for maximizing distance?
- What speed was the ball traveling upon impact, and what force would be necessary to catch it?
- The video mentions a historical throw by Bomar that reached 35 yards horizontally. How does this compare to Manning's attempt?
- Lastly, there’s a claim that the ball travels a total distance of 112 yards, which raises questions about what that entails.
Of course, we could also tackle the motion of the ball considering air resistance, but that adds a layer of complexity (which I intend to explore nonetheless).
Section 1.1: Basic Principles of Projectile Motion
To understand the throw, we start with basic projectile motion. When a ball is launched with an initial velocity ( v_0 ) at an angle ( theta ) from a height ( h ), the only force acting on it after launch is gravity. This scenario defines it as projectile motion. If other forces, like air resistance, are present, it no longer qualifies as such.
The gravitational force operates solely in the vertical direction, resulting in a constant horizontal velocity. The vertical velocity, however, experiences a constant acceleration of (-g) (where ( g = 9.8 , m/s² )). This leads us to two key equations of motion.
Both equations depend on the initial velocities in either the x or y direction. Using the initial launch angle, we can calculate these velocities. We already know the initial vertical position (height ( h )) and the final position (ground level, or 0). The only unknowns are the final horizontal position ( x ) and the time of flight, which we can determine using the vertical kinematic equation.
Yes, this involves using the quadratic formula. This will yield two time values, but we only need the positive one. We can then apply this time to the horizontal motion to find how far the ball travels before hitting the ground.
To be thorough, let’s substitute the expressions for the initial x and y velocities into our equations.
Section 1.2: Determining the Building Height
We know the height of the Bryant Park Hotel is 103 meters, but this measurement is to the roof. Clearly, Manning isn't throwing from the very top. How high is he for this throw?
Using an image for scale, I can estimate the height from the balcony. Employing Tracker Video Analysis, I found a height of approximately 88 meters. While this is still an estimate, it provides a starting point for further calculations.
Chapter 2: Analyzing Launch Angles and Distances
With an approximate launch speed of 60 mph (27 m/s), I can experiment with various launch angles to see how far the ball travels. This exploration reveals that the maximum distance is achieved at a launch angle of 29 degrees—not the commonly assumed 45 degrees. At this optimal angle, the ball could cover a distance of 138 meters (151 yards), exceeding the 85-yard target mentioned in the video.
The first video showcases Peyton Manning throwing a football off a skyscraper, demonstrating the physics involved in this remarkable feat.
What about the 112 yards mentioned in the video? Let's examine the scenario. The 85 yards represent the ground distance to the receiver, but Peyton suggests the ball travels 112 yards through the air. Is this the total path length? Perhaps it's the combined distance traveled upward and downward, but it remains unclear.
To clarify this, I can model the football's motion over short time intervals, adding up the distances traveled. I plan to implement this in Python to visualize the results.
The second video tackles the question of how far Peyton Manning can throw a football from a building, providing further insight into the physics behind this impressive throw.
Section 2.1: Considering Air Resistance
A real football's motion cannot be categorized purely as projectile motion, as it is affected by both gravity and air resistance. The air resistance can be modeled using various parameters, including air density, cross-sectional area, and the drag coefficient.
Incorporating air resistance complicates the problem, as the acceleration is no longer constant. To accurately model this, we must use numerical methods, breaking the motion into short time intervals and calculating the changes in velocity and position incrementally.
By applying this method to Peyton's throw from the 23rd floor, we can analyze how launch velocity and angle affect the range. Preliminary findings indicate that the maximum range occurs at a launch angle between 30 and 35 degrees—not 45 degrees. Even at a speed of 25 m/s, the ball reaches a distance of 113 meters (124 yards), which is sufficient for Manning to hit Carter.
However, no calculated distance matches the stated 112 yards. This discrepancy raises further questions about its origin and significance.
Homework
There's still plenty of exploration left for you. Here are some questions to ponder:
- Estimate the distance over which Chris Carter catches the ball and calculate the impact force. Would it be painful to catch?
- Determine the terminal velocity of a football. How high would it need to be thrown to reach terminal velocity before impact?
- Modify the code to account for instances when the ball transitions from a spiral to a flat spin.
- Use video analysis to estimate the launch velocity and angle of Manning's throw.
- Measure the flight time and final velocity of the ball in the video. Do these figures align with your numerical model?
- If Manning can throw a ball 50 yards on the field, what launch velocity and angle would be necessary, using the same drag coefficient parameters as above?