JEE Physics 2025 Question with Solution

Given: A cup of coffee cools from $90^\circ C$ to $80^\circ C$ in $t$ minutes, and the surrounding temperature is $20^\circ C$.

Find: The time taken for a similar cup to cool from $80^\circ C$ to $60^\circ C$ at the same surrounding temperature.

Using Newton's law of cooling in integrated form:

$$\frac{dT}{dt} = -k(T-T_s)$$

Integrating between temperatures $T_1$ and $T_2$ gives

$$\int_{T_1}^{T_2} \frac{dT}{T-T_s} = -k \int_0^t dt$$

so,

$$\ln \frac{T_2-T_s}{T_1-T_s} = -kt$$

Hence,

$$k = \frac{1}{t} \ln \frac{T_1-T_s}{T_2-T_s}$$

For cooling from $90^\circ C$ to $80^\circ C$,

$$k = \frac{1}{t} \ln \frac{90-20}{80-20} = \frac{1}{t} \ln \frac{70}{60} = \frac{1}{t} \ln \frac{7}{6}$$

Let the required time for cooling from $80^\circ C$ to $60^\circ C$ be $t_2$. Then,

$$k = \frac{1}{t_2} \ln \frac{80-20}{60-20} = \frac{1}{t_2} \ln \frac{60}{40} = \frac{1}{t_2} \ln \frac{3}{2}$$

Since $k$ is the same in both cases,

$$\frac{1}{t} \ln \frac{7}{6} = \frac{1}{t_2} \ln \frac{3}{2}$$

Therefore,

$$t_2 = t \frac{\ln(3/2)}{\ln(7/6)}$$

Using the approximation given in the solution,

$$\ln(3/2) = 0.405, \quad \ln(7/6) = 0.154$$

So,

$$\frac{\ln(3/2)}{\ln(7/6)} = \frac{0.405}{0.154} \approx 2.63 \approx \frac{13}{5}$$

Hence,

$$t_2 = \frac{13}{5} t$$

Therefore, the time taken is $\frac{13}{5} t$, so the correct option is A.

Given: The cup cools from $90^\circ C$ to $80^\circ C$ in $t$ minutes, and the room temperature is $20^\circ C$.

Find: The time for cooling from $80^\circ C$ to $60^\circ C$.

Using the average form of Newton's law of cooling:

$$\frac{90-80}{t} = k\left(\frac{90+80-20}{2}\right)$$

and

$$\frac{80-60}{t'} = k\left(\frac{80+60-20}{2}\right)$$

From these two equations,

$$\frac{10}{t} = 75k$$

and

$$\frac{20}{t'} = 60k$$

Eliminating $k$,

$$10t' = 26t$$

Thus,

$$t' = \frac{13}{5} t$$

This shortcut works because over a small temperature interval, the cooling rate may be estimated using the average excess temperature above the surroundings. Therefore, the correct option is A.