---
title: "Physical vs mathematical models"
subject: "Simulation and Modeling"
chapter: 2
key: "semester_8/simulation_and_modeling:physical-vs-mathematical-model"
source: https://www.bctengineer.com/q/semester_8/simulation_and_modeling/physical-vs-mathematical-model
site: BCT Engineering
---

# Physical vs mathematical models

Simulation and Modeling · Chapter 2: Physical and Mathematical models

- Asked in: 2070 Magh, 6 marks
- Video: [Modeling Physical Systems, An Overview](https://www.youtube.com/watch?v=580A8gNFOdk) — Brian Douglas
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_8/simulation_and_modeling/physical-vs-mathematical-model
- Whole chapter: https://www.bctengineer.com/md/semester_8/simulation_and_modeling/chapter-2

## Question

Differentiate between physical and mathematical models with proper examples.

## Short answer

A physical model is a tangible, scaled hardware version (e.g., wind tunnel aircraft), whereas a mathematical model uses symbolic equations (e.g., M x'' + D x' + K x = F(t)). Physical models are costly and hard to modify; mathematical models are flexible and cheap to test.

## Answer

A **physical model** is a tangible, scaled or analogous version of the system built from real materials. A **mathematical model** represents the system by symbols, equations and logical relationships.

| Point | Physical model | Mathematical model |
|---|---|---|
| Representation | Real hardware (scale or analogue) | Equations and logic |
| Form | Tangible, visible | Abstract, symbolic |
| Cost of building | Usually high | Usually low |
| Modification | Difficult, slow | Easy, change parameters in the equations |
| Flexibility | Limited | Highly flexible |
| Accuracy | Real effects included automatically | Depends on assumptions made |
| Time to study | Often real or scaled time | Can be compressed or expanded on computer |
| Solution | Observation and measurement | Analytical or numerical solution, simulation |
| Typical use | Testing in realistic conditions, demonstration | Prediction, optimization, large systems |

#### Examples of physical models
- A scale model of an aircraft in a wind tunnel.
- A ship model in a water tank.
- A model of a dam or a bridge.
- A flight simulator cockpit.
- A mechanical spring-mass-damper rig for a car suspension.

#### Examples of mathematical models
- `F = m a` for a moving body.
- Suspension: `M x'' + D x' + K x = F(t)`.
- Queue: `L = rho/(1 - rho)` for an M/M/1 queue.
- Population: `dP/dt = k P`.

Physical models are used when the physics is too complex to write down or when realistic demonstration is needed. Mathematical models are used when many alternatives must be tested cheaply. Often both are used: a mathematical model is validated by results from a physical model.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which is a difference between mathematical and physical models?
   A. Physical models are easier and cheaper to modify
   B. Both are equally costly to change
   C. Mathematical models cannot be solved by computer
   D. Mathematical models are easy to modify by changing parameters, physical ones are slow
   Answer: D. Changing parameters in equations is easy; physical models are costly and slow to change. (Key point: Modification ease)

2. Why can a physical model be more realistic?
   A. Real effects are included automatically, with fewer assumptions
   B. It always runs faster than real time
   C. It is symbolic
   D. It needs no measurement
   Answer: A. A mathematical model's accuracy depends on its assumptions, while a physical model includes real effects automatically. (Key point: Accuracy)

3. Which is a physical model?
   A. F = m a
   B. dP/dt = k P
   C. A ship model in a water tank
   D. L = rho/(1 - rho)
   Answer: C. A ship model in a tank is tangible hardware; the others are equations. (Key point: Example classification)

## Related in Simulation and Modeling

- [Static vs dynamic mathematical model](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Astatic-vs-dynamic-math-model) — Simulation and Modeling, chapter 2, asked 3×

## Related in other subjects

- [Proper vs improper handoff; cellular concept N = 7](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Ahandoff-situation-cellular-n7) — Wireless Communication, chapter 2
- [Physical vs logical clock; sync method](https://www.bctengineer.com/question.md?key=semester_7%2Fdistributed_system%3Aphysical-clock-sync-difficulty) — Distributed System, chapter 5, asked 2×
- [Physical vs logical clocks; Lamport](https://www.bctengineer.com/question.md?key=semester_7%2Fdistributed_system%3Alamport-physical-compare) — Distributed System, chapter 5
- [Mathematical modelling and applications of FEM](https://www.bctengineer.com/question.md?key=semester_8%2Ffinite_element_method%3Amathematical-modeling-fem-applications) — Finite Element Method, chapter 1