Newton’s Law of Cooling
Temperature approaches ambient exponentially.
Key idea
Temperature approaches ambient exponentially.
In depth
The rate of temperature change is proportional to the difference from the surroundings: T′ = −k(T − Tₛ). The solution is T = Tₛ + (T₀ − Tₛ)e^(−kt), approaching room temperature exponentially.
Worked example
Coffee at 90 °C in a 20 °C room, k = 0.1/min: T(10) = 20 + 70e⁻¹ ≈ 45.8 °C.
Formula
Real-world uses
- Estimating the time of death in forensic work or when food is safe to refrigerate
Common mistake
Using T instead of the difference T − Tₛ.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Newton’s Law of Cooling?
Temperature approaches ambient exponentially. It is a topic at the advanced level in Differential Equations, Abstract Algebra & Applied Math.
What is the formula for Newton’s Law of Cooling?
The key formula is T = Tₐ + (T₀−Tₐ)e^(−kt).
Where is Newton’s Law of Cooling used in real life?
Real-world examples: Estimating the time of death in forensic work or when food is safe to refrigerate.
What should I learn before Newton’s Law of Cooling?
Make sure you are comfortable with Logistic Growth, the previous topic in Differential Equations, Abstract Algebra & Applied Math.
What is a common mistake with Newton’s Law of Cooling?
Using T instead of the difference T − Tₛ.
How can I practise Newton’s Law of Cooling?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Newton’s Law of Cooling?
Move on to Mixing Problems next.
Related advanced topics in Differential Equations, Abstract Algebra & Applied Math
- What Is a Differential Equation? – An equation relating a function and its derivatives.
- Order and Linearity of ODEs – Classify equations by highest derivative and linearity.
- Direction Fields – Slope fields show solution curves visually.
- Separable Equations – Separate variables and integrate both sides.
- First-Order Linear Equations – Use an integrating factor.
- Exponential Growth and Decay Models – dy/dt = ky gives y = y₀eᵏᵗ.