Sling Shot Physics: Can You Really Launch a Human Between Buildings?
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Chapter 1: The Internet's Wild World of Slingshots
Have you ever browsed the vast expanse of "The Internet"? If so, you've likely stumbled upon a plethora of videos—some absurd, some genuine, and others downright fabricated. One such video features an individual in a colossal slingshot atop a skyscraper. With a dramatic pullback, the slingshot releases, sending the person soaring towards another building with a massive net to catch him. While the scene appears thrilling, fear not—this is merely a clever fabrication from an old advertisement.
Despite its fictional nature, we can still extract some physics from this idea. If such a stunt were feasible, I doubt it would actually succeed. A minor adjustment in launch velocity could easily result in missing the safety net entirely.
We can pose several intriguing physics questions here:
- What launch speed would be necessary to reach the adjacent building?
- How can we estimate the spring constant of the slingshot launcher?
- What kind of acceleration would a human experience during launch, and would it be safe?
- How about the deceleration upon landing?
- Is it reasonable to overlook air resistance in this scenario?
- If the launch angle varies slightly, how often might the human miss the target?
- What costs would be associated with such a leap, and what type of insurance would be required?
- Finally, if you were to survive this leap, would you earn a commemorative t-shirt?
While I won't address every question, I'll provide a foundation to get you started. The rest is up to your imagination (and some homework).
Video Description: Witness the exhilarating experience of a human slingshot in Dubai, showcasing the thrill and daring of this extreme activity.
Estimating Launch Velocity
Forget about the launcher for a moment. Once the person is propelled into the air, we need to determine the speed necessary to reach the other building. I'll begin with a few assumptions:
- The human is launched at a 35-degree angle (a rough estimate).
- Air resistance is negligible.
- The starting and landing points are at the same elevation.
- The distance between buildings is 450 meters (as determined via Google Maps).
With these parameters, we have a straightforward projectile motion problem. In the absence of air resistance, we can analyze this as two independent motions: horizontal movement at a constant velocity and vertical motion affected by gravity.
Using basic kinematic equations, we can express both the horizontal (x) and vertical (y) components. By setting our coordinate system at the launch point (x_i = 0, y_i = 0) and noting that our final position x_f is 450 meters with a final y position of 0 meters, we simplify our calculations.
By substituting values into the equations, we find that the necessary launch speed is approximately 68.5 meters per second (about 153 mph).
Bungee Spring Constant Estimation
Next, let's determine the spring constant of the bungee used to launch our daring participant. Starting with some assumptions:
- The bungee cord adheres to Hooke's Law, meaning the force it exerts is proportional to its stretch.
- Its unstretched length is zero.
- The bungee is stretched by 10 meters, and the combined mass of the human and chair is 100 kg.
Using the work-energy principle, we can define the kinetic and potential energy terms and establish the work-energy equation. This results in a spring constant of approximately 4693 Newtons per meter—a notably high value for a bungee, affirming the stunt's fictional nature.
Acceleration During Launch
Humans can endure only so much acceleration before injuries occur. While falling, the rapid deceleration can lead to severe consequences. Fortunately, research provides insights into the maximum tolerable acceleration (in g's) for humans.
When launched via bungee, a person could theoretically withstand about 35 g's for a brief moment (equivalent to 343 m/s²). We can calculate this in two ways: first, using the spring constant, and second, through video analysis of the launch.
Calculating using the spring constant yields an initial acceleration of 469 m/s², which decreases as the launch proceeds. Averaging this gives about 235 m/s², which is still alarming. The video analysis, however, indicates an average acceleration of around 257 m/s², surprisingly consistent with our previous calculations.
Considering Air Resistance
Is it reasonable to disregard air resistance in our calculations? To understand the impact, we can compare trajectories with and without air drag. By modeling the interaction between the human and the air, we can visualize how air resistance alters the trajectory.
While detailing the calculations is complex, a numerical approach can simplify the process. However, determining the parameters for a human's drag force requires knowledge of variables such as surface area and drag coefficient. We can find these by observing that a skydiver's terminal velocity is about 120 mph (54 m/s).
After running simulations, we observe a significant difference in outcomes with air resistance compared to without. If we want a human to land safely, we'd need a launch velocity of 146.5 m/s, which is considerably higher than the earlier estimate.
Homework Assignment
Here's your chance to engage with the material. Complete the following tasks for credit:
- Recalculate the spring constant and acceleration using the adjusted launch velocity that accounts for air resistance.
- Theoretical calculations suggest a launch time of 10 seconds with air resistance, while the video indicates around 6 seconds. Can you adjust the launch angle to reconcile these times?
- If the slingshot launch cart is pulled back by just 5 centimeters, how might that slight lateral component affect the landing accuracy?
- At the end of the video, the launch subject lands in a pool. Estimate the pool's depth necessary for a safe landing (or as safe as it can be).
Video Description: This video showcases a team transforming a trampoline into a thrilling human slingshot, demonstrating creativity and engineering in action.