Limits of ideal mechanical models

Ideal mechanical models rest on conservation laws that hold only under certain conditions: the system is closed, the bodies are treated as point masses, and dissipative forces are absent.

Momentum conservation follows from Newton's second and third laws: by superposition, internal forces cancel in pairs, while external forces give the impulse $\vec{F}_{\text{ext}}\,\Delta t = \Delta \vec{p}$. Therefore momentum is conserved only up to the dimensionless parameter

$$\varepsilon_p = \frac{F_{\text{ext}}\,\Delta t}{p}.$$

Mechanical energy is conserved only when the work of non-conservative forces (friction, air drag, inelastic impact) is small compared with the kinetic energy: $\varepsilon_E = \dfrac{|A_{\text{nc}}|}{E_k} \ll 1$.

The limit of the model is not a qualitative "yes or no" but a number: for $\varepsilon \lesssim 10^{-2}$ the error of the ideal calculation stays within a few percent, while for $\varepsilon \gtrsim 1$ the ideal model fails and external forces and dissipation must be included explicitly.

A cart $m_1 = 0{,}5$ kg moves at $v_1 = 2{,}0$ m/s along a horizontal rail and couples with a stationary cart $m_2 = 0{,}5$ kg. The friction force on the carts is $F_{\text{fr}} = 0{,}5$ N, and the coupling lasts $\Delta t = 0{,}02$ s. Check whether momentum conservation applies and find the speed of the carts after coupling.

A bullet of mass $m = 10$ g flies at $v = 400$ m/s and embeds itself in a hanging block of mass $M = 1{,}99$ kg. The impact lasts $\Delta t \approx 10^{-3}$ s. (a) Find the block's speed just after the impact and the height it rises. (b) Check the applicability of momentum and energy conservation at each stage.

A ball of mass $m = 0{,}40$ kg and radius $r = 3{,}7$ cm is thrown with speed $v_0$. Estimate the largest $v_0$ for which the ideal drag-free flight model gives an error below 1%. Use $F_{\text{drag}} = \frac12 C_d \rho A v^2$ with $C_d = 0{,}47$, $\rho = 1{,}2$ kg/m³, $A = \pi r^2$. Explain why the value obtained is only a scaling estimate.

Run a numerical experiment in a simulation (e.g., PhET "Collision Lab" or "Energy Skate Park") where elasticity, friction, and interaction time can be varied independently.

1. For each parameter set, record the momentum and kinetic energy of the system before and after the interaction.
2. Fill in a table: $\varepsilon_p = \dfrac{|p_{\text{final}} - p_{\text{initial}}|}{p_{\text{initial}}}$, $\varepsilon_E = \dfrac{|E_{\text{final}} - E_{\text{initial}}|}{E_{\text{initial}}}$.
3. Find the parameter thresholds at which $\varepsilon$ exceeds 1% and 10%, and state which model (ideal or non-ideal) works in each region.

The simulation is preferred because precise measurement of a short impact with household instruments gives large random errors. If you do drop a light ball at home, do it away from furniture and never use heavy objects.

Show that the limit of the ideal model is governed by the dimensionless parameter $\varepsilon$, and indicate where the 1% error threshold lies.

A combined sketch: on the left, a diagram of the system with internal forces (paired opposite arrows) and external forces (friction, gravity, air drag); on the right, a log-log graph with $\varepsilon$ on the horizontal axis and the relative error of the ideal model on the vertical axis, marking the vertical boundary $\varepsilon \approx 10^{-2}$ and three regions: "ideal model", "intermediate", "non-ideal model".

A ball of mass $m=0{,}62\ \mathrm{kg}$ falls from height $h_1=1{,}8\ \mathrm{m}$ and after the bounce rises to $h_2=1{,}2\ \mathrm{m}$. With $g=9{,}8\ \mathrm{m/s^2}$, the initial potential energy is $E_{p1}=mgh_1\approx 10{,}9\ \mathrm{J}$, and after the bounce $E_{p2}=mgh_2\approx 7{,}3\ \mathrm{J}$; the difference $\Delta E\approx 3{,}6\ \mathrm{J}$ goes into heating the ball and floor and into sound.

This tests the ideal assumption of conservation of mechanical energy: if the impact were perfectly elastic and dissipation-free, $E_{p2}$ would equal $E_{p1}$. Real deformation and internal friction show the limit of the conservative-system model.

A car of mass $m=1200\ \mathrm{kg}$ moves at speed $v=20\ \mathrm{m/s}$ ($72\ \mathrm{km/h}$) and stops over distance $s=40\ \mathrm{m}$. Its kinetic energy is $E_k=\frac12 mv^2=240\ \mathrm{kJ}$; the average braking force is $F=E_k/s=6000\ \mathrm{N}$. This energy does not disappear but turns into heat in the brake discs and tyres.

The ideal model of an isolated system with no external forces does not apply: friction from the road is external, so the car's momentum is not conserved and mechanical energy is not conserved. We use the energy-change law with work done by external forces, $A_{\text{ext}}=\Delta E_k$.

A phone of mass $m=0{,}18\ \mathrm{kg}$ falls from height $h=1{,}2\ \mathrm{m}$. Before impact $v=\sqrt{2gh}\approx 4{,}85\ \mathrm{m/s}$, and its kinetic energy is $E_k\approx 2{,}1\ \mathrm{J}$. Stopping on the floor over $d_1=1\ \mathrm{mm}$ gives an average impact force $F_1\approx E_k/d_1\approx 2{,}1\ \mathrm{kN}$; on a mattress over $d_2=5\ \mathrm{cm}$ the force is $F_2\approx E_k/d_2\approx 42\ \mathrm{N}$.

The model of a perfectly rigid body with an instantaneous impact and conservation of mechanical energy is limited here: a real impact lasts a finite time, the body and support deform, part of the energy goes into sound and heat, and the impact force depends on the stopping distance. This exercises impulse-energy analysis instead of an ideal elastic collision.