A.2.3 What is Mathematical Modelling – Class 11 Mathematics

Introduction

Mathematical Modelling is the art and science of translating real-world situations into mathematical language. It involves using equations, graphs, and logical reasoning to represent complex systems such as population growth, projectile motion, heat transfer, or economic trends. For JEE aspirants, understanding mathematical modelling builds strong problem-solving and analytical thinking skills essential in both Mathematics and Physics.

A mathematical model bridges the gap between the real world and the mathematical world. It provides a simplified but effective description of how systems behave, allowing predictions, analysis, and optimization.

Definition

A mathematical model is a mathematical representation (such as an equation or inequality) that describes a real-world situation. It involves:

• Identifying the problem
• Formulating it mathematically
• Solving the resulting equations
• Interpreting the solution in real-world terms

In simple terms, mathematical modelling helps us answer “What happens if…?” questions using mathematics.

Purpose of Mathematical Modelling

The goal of mathematical modelling is to make sense of reality using mathematics. It helps to:

1. Predict future outcomes (e.g., population growth, motion of a projectile).
2. Understand underlying mechanisms (e.g., rate of cooling, spread of disease).
3. Optimize performance (e.g., minimizing cost, maximizing profit).
4. Test hypotheses and refine them using data.

In the JEE context, this understanding aids in applying mathematical principles to engineering, physics, and data analysis.

Steps in the Mathematical Modelling Process

A systematic process ensures clarity and precision in developing a model:

1. Problem Identification

Define what is being studied. Example: How does a ball move under gravity? How does a bacteria population change over time?

2. Simplifying Assumptions

Simplify complex reality. Example: Ignore air resistance or assume uniform gravity.

3. Defining Variables and Parameters

Identify measurable quantities like time (t), position (x), or growth rate (k).

4. Constructing Mathematical Relationships

Build equations that connect variables, e.g. v = u + gt or P(t) = P₀e(kt).

5. Solving the Model

Use algebraic, differential, or numerical methods to find solutions.

6. Interpreting the Results

Translate mathematical outcomes into real-world meaning.

7. Validating and Refining the Model

Compare with experimental or observed data to confirm accuracy.

Example 1: Projectile Motion

A body thrown with velocity u at an angle θ follows a parabolic path. The equations of motion are:

x = u cosθ * t
y = u sinθ * t – (1/2) g t²

This model helps predict range, time of flight, and maximum height — key concepts in JEE Physics.

Example 2: Exponential Population Growth

Population growth proportional to its current size can be written as:

dP/dt = kP

Solving gives:

P(t) = P₀ e(kt)

This model helps analyze biological and environmental growth phenomena.

Example 3: Cooling of a Hot Body

According to Newton’s Law of Cooling:

dT/dt = -k (T – Tₐ)

Solving gives:

T(t) = Tₐ + (T₀ – Tₐ)e(-kt)

This equation predicts how quickly an object cools in a surrounding environment.

Types of Mathematical Models

Mathematical models can be categorized based on their nature:

Importance in JEE Preparation

Mathematical modelling integrates multiple JEE topics such as:

• Kinematics and Calculus – for motion equations and rate of change.
• Differential Equations – for growth and decay processes.
• Optimization Problems – for maxima and minima in applied scenarios.
• Probability Models – for predicting uncertain outcomes.

It enhances logical problem-solving by helping students identify what kind of mathematics applies to what type of real-world problem.

Marks Weightage in JEE

Advantages of Mathematical Modelling

1. Provides quantitative understanding of real phenomena.
2. Enables prediction and control of processes.
3. Simplifies complex systems into manageable equations.
4. Allows simulation and optimization before real implementation.
5. Bridges theoretical and experimental studies.

Limitations of Mathematical Modelling

1. Simplified assumptions may reduce accuracy.
2. Errors in data can mislead predictions.
3. Non-linear or chaotic systems may be difficult to model.
4. Validation issues arise when data is insufficient.

JEE-Based Problem Example

Problem: 

The temperature of an object decreases according to Newton’s Law of Cooling. If its initial temperature is 80°C and surrounding temperature is 30°C, find the time when the temperature becomes 50°C, given k = 0.1 min⁻¹.

Solution:

T(t) = 30 + (80 – 30)e(-0.1t)
50 = 30 + 50e(-0.1t)
→ e(-0.1t) = 0.4 → t = (ln(1/0.4))/0.1 ≈ 9.16 minutes.

FAQs

Q1. What is mathematical modelling in simple terms?

It is the process of describing real-world problems using mathematical equations and logic.

Q2. Why is it important for JEE aspirants?

It connects physics, mathematics, and real-world reasoning — essential for solving application-based problems.

Q3. What are the stages in modelling?

Defining the problem, simplifying, forming equations, solving, interpreting, and validating.

Q4. What mathematical tools are used in modelling?

Calculus, algebra, geometry, statistics, and differential equations.

Q5. Can all real-world phenomena be modelled mathematically?

Not perfectly, but approximate models can still provide valuable insight and predictions.

Conclusion

Mathematical Modelling is a cornerstone of applied mathematics that transforms real-life problems into solvable mathematical structures. It builds analytical thinking, precision, and creativity — essential traits for JEE aspirants. Whether it’s the motion of a particle or the spread of heat, models reveal the underlying mathematical beauty of nature and technology. Mastering this concept prepares students to tackle interdisciplinary challenges confidently in engineering and beyond.