Manipulate mass, cord length, spring constant, and damping coefficient to simulate bungee jumping via Runge-Kutta numerical integration. Visualize lowest point, oscillation period, and energy conversion in real time.
Physics of the Bungee Cord — Elastic Potential Energy & Free Fall
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Professor, in bungee jumping, after you jump, it's just normal free fall until the cord tightens, right? What happens when the cord starts to stretch?
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That's right, initially it's just free fall. When the cord stretches beyond its natural length $L_0$, a spring force $F = k(x - L_0)$ acts upward. The equation of motion is $m\ddot{x} = mg - k(x - L_0) - b\dot{x}$ (with downward as positive). $b\dot{x}$ is the damping force from air resistance and internal losses.
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I see. What happens at the lowest point? That's the moment when velocity becomes zero, right?
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Exactly. At the lowest point, velocity is zero, so using energy conservation we get $x_{max} = L_0 + mg/k + \sqrt{(mg/k)^2 + 2mgL_0/k}$. For example, with m=70kg, k=40 N/m, L₀=20m, $mg/k = 70 \times 9.81 / 40 \approx 17.2\ \text{m}$, $x_{max} \approx 20 + 17.2 + 21.3 \approx 58.5\ \text{m}$. The cord stretches to about three times its length.
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Three times! How much G-force do you feel at the lowest point?
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The acceleration at the lowest point is when the cord force is maximum. $a = k(x_{max} - L_0)/m - g$, and in the example above it's about $(40 \times 38.5)/70 - 9.81 \approx 22 - 9.81 \approx 12.2\ \text{m/s}^2 \approx 1.2G$. The stiffer the cord, the higher the G-force; typical bungee jumps experience 2–3G. Considering a car brake is about 0.8G and fighter pilots can withstand up to 9G, bungee G-forces are actually not that large.
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That's lower than I expected. If damping were zero, would it oscillate forever? Where does the energy actually get lost?
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Ideally, it would oscillate forever. In reality, damping comes from ① internal friction in the cord (viscoelasticity of rubber/latex), ② air resistance, and ③ tiny energy dissipation at the attachment point. You can verify in the simulator by setting the damping coefficient b to zero—the oscillation continues. Also, the 'Energy Conversion' tab shows the transitions between potential energy, kinetic energy, and elastic energy.
Frequently Asked Questions
How is the lowest point of a bungee jump calculated?
For the undamped case, we use energy conservation. When falling from the platform (reference point) to the lowest point $x_{max}$, the decrease in gravitational potential energy equals the increase in elastic potential energy of the cord: $mgx_{max} = \frac{1}{2}k(x_{max} - L_0)^2$ Solving this quadratic equation gives $x_{max} = L_0 + mg/k + \sqrt{(mg/k)^2 + 2mgL_0/k}$.
What happens if the spring constant k is large?
The stiffer the cord (larger k), the shallower the lowest point (less stretch), the shorter the oscillation period $T \approx 2\pi\sqrt{m/k}$ (faster oscillation), and the higher the G-force at the bottom. Conversely, a smaller k means the cord stretches more, the lowest point is deeper, and the risk of ground impact increases. In bungee design, adjusting k to the jumper's weight is key to safety.
What is the effect of the damping coefficient?
A larger damping coefficient $b$ (unit N·s/m) makes oscillations decay faster. $b = 0$ gives perfect elasticity (perpetual oscillation), $b = 2\sqrt{mk}$ gives critical damping (slowly settles to equilibrium without overshoot), and higher values give overdamping. The equivalent damping coefficient of a real bungee cord (natural rubber) is on the order of a few N·s/m. Air resistance is proportional to the square of velocity, but at low speeds it can be approximated as $b\dot{x}$.
What material is a real bungee cord made of?
It consists of bundles of natural latex or synthetic rubber ropes, which can stretch to 2–3.5 times their natural length. They exhibit nonlinear elasticity beyond Hooke's law (linear spring), but safety design uses a linear approximation with a margin. The safety factor is typically 3–5 (cord breaking strength / design maximum load). For a 65 kg person, the spring constant is roughly 30–60 N/m, but it is designed based on the platform height and minimum allowable height.
What is the Runge-Kutta method? Why is it better than the Euler method?
The 4th-order Runge-Kutta method (RK4) evaluates the slope four times per step and uses a weighted average to estimate the next value. Accuracy improves with the 5th power of the step size $h$ (error $O(h^5)$). The simple Euler method has error $O(h^2)$ and can produce unstable solutions with increasing energy in nonlinear oscillations like bungee jumping. RK4 is several orders of magnitude more accurate than Euler for the same step count and is the standard method for physical simulations.
What is Bungee Jump Physics?
Bungee Jump Physics is a fundamental topic in engineering and applied physics. This interactive simulator lets you explore the key behaviors and relationships by directly manipulating parameters and observing real-time results.
By combining numerical computation with visual feedback, the simulator bridges the gap between abstract theory and physical intuition — making it an effective learning tool for students and a rapid-verification tool for practicing engineers.
Physical Model & Key Equations
The simulator is based on the governing equations behind Bungee Jump Physics Simulator. Understanding these equations is key to interpreting the results correctly.
Each parameter in the equations corresponds to a slider in the control panel. Moving a slider changes the equation's solution in real time, helping you build a direct connection between mathematical expressions and physical behavior.
Real-World Applications
Engineering Design: The concepts behind Bungee Jump Physics Simulator are applied across mechanical, structural, electrical, and fluid engineering disciplines. This tool provides a quick way to estimate design parameters and sensitivity before committing to full CAE analysis.
Education & Research: Widely used in engineering curricula to connect theory with numerical computation. Also serves as a first-pass validation tool in research settings.
CAE Workflow Integration: Before running finite element (FEM) or computational fluid dynamics (CFD) simulations, engineers use simplified models like this to establish physical scale, identify dominant parameters, and define realistic boundary conditions.
Common Misconceptions and Points of Caution
Model assumptions: The mathematical model used here relies on simplifying assumptions such as linearity, homogeneity, and isotropy. Always verify that your real system satisfies these assumptions before applying results directly to design decisions.
Units and scale: Many calculation errors arise from unit conversion mistakes or order-of-magnitude errors. Pay close attention to the units shown next to each parameter input.
Validating results: Always sanity-check simulator output against physical intuition or hand calculations. If a result seems unexpected, review your input parameters or verify with an independent method.