Mathematics · Differential Equations
Newton's Law of Cooling Calculator
Estimate an object's temperature as it approaches a constant ambient temperature.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- Initial difference = 90−20=70.
- Decay factor = e^(−0.1×10)=0.36787944117144233.
- T(10)=20+70×0.36787944117144233=45.75156088200096.
Problem → model → reason → result
What problem does this model solve?
Estimate an object's temperature as it approaches a constant ambient temperature.
Why does the model apply?
The temperature changes at a rate proportional to its difference from the surroundings, so that difference decays exponentially.
What assumptions does it make?
The displayed differential equation represents the system, coefficients remain constant over the chosen interval, and the supplied initial condition selects one solution.
Formula
T(t)=Tₐ+(T₀−Tₐ)e⁻ᵏᵗ
Calculation and working
The calculator above substitutes your inputs into the model and leaves the calculation steps visible so the answer can be checked rather than merely accepted.
What does the result mean?
The result answers the stated problem in the units implied by your inputs. Read its sign, size and units together, then compare it with the original values before drawing a conclusion.
Worked example
An object at 90°C in a 20°C room with k=0.1 after 10 minutes is about 45.75°C.
Common mistake
The model approaches ambient temperature; it does not normally cross it.
When does this model not apply?
These closed-form and one-step numerical models do not capture changing coefficients, discontinuities, measurement uncertainty or every nonlinear system.
How to learn with this calculator
Begin with the worked example, then change one value while keeping the others fixed. Compare the new result and calculation steps to identify which part of the formula changed.
Clear answers
Frequently asked questions
What does the Newton cooling do?
Estimate an object's temperature as it approaches a constant ambient temperature.
How does the Newton cooling work?
The calculator applies T(t)=Tₐ+(T₀−Tₐ)e⁻ᵏᵗ. The temperature changes at a rate proportional to its difference from the surroundings, so that difference decays exponentially.
What can I learn from the Newton cooling?
It connects the mathematical rule to your chosen numbers and shows each calculation step. Change one input at a time to see how the result responds.
Does MW SysArc receive or store what I enter?
No. The calculation runs locally in your browser. MW SysArc does not receive or store your calculation inputs.
How should I use the result?
Use the steps to understand the method, then verify important school or professional work using the notation and rounding rules required in your setting.
Last reviewed 2026-07-14. Calculations tested 2026-07-14.