Two identical metal bars are heated in two different temperature and allowed to cool in the same surroundings. Which one of the following figures correctly shows their cooling curves?

According to Newton's law of cooling, the rate of cooling of a body is directly proportional to the temperature difference between the body and its surroundings.

The temperature $T$ of the body at any time $t$ is given by the equation:
$T = T_0 + (T_i - T_0)e^{-kt}$
where $T_0$ is the temperature of the surroundings, $T_i$ is the initial temperature of the body, and $k$ is a positive constant that depends on the nature and area of the surface.

Since the two metal bars are identical, the cooling constant $k$ is the same for both.

Let the initial temperatures of the two bars be $T_{i1}$ and $T_{i2}$. The temperature difference between them at any time $t$ is:
$\Delta T = T_1 - T_2 = (T_{i1} - T_{i2})e^{-kt}$

Since $T_{i1} \neq T_{i2}$, the temperature difference $\Delta T$ will never be zero for any finite time $t$. This means the cooling curves will never intersect.

Furthermore, as $t \to \infty$, both $e^{-kt} \to 0$, so the temperatures of both bars will exponentially approach the same surrounding temperature $T_0$.

The graph in option (D) correctly represents two non-intersecting exponential decay curves that approach the same horizontal asymptote.

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