Animated Solution for Physics - Thermodynamics: Cooking gas containers are kept in a lorry moving with uniform speed. The temperature of the gas molecules inside will

Imagine you are standing on the side of a highway, watching a massive lorry cruise past you at a perfectly uniform speed. Inside the cargo hold of this lorry sits a sealed container filled with cooking gas. A fascinating question arises: Does the fact that the lorry is moving at $80\text{ km/hr}$ mean the gas inside is hotter than if the lorry were parked?

To answer this, we must dive into the microscopic world and understand what temperature truly is.

In our everyday experience, we associate kinetic energy with speed. A fast-moving car has more kinetic energy than a slow one. However, temperature is not a measure of the macroscopic kinetic energy of an entire object.

Instead, temperature is a measure of the average random translational kinetic energy of the molecules, measured strictly with respect to the center of mass of the gas.

Mathematically, the random kinetic energy is given by:

Here, ${\vec{v}}_i$ is the velocity of an individual molecule, and ${\vec{v}}_{cm}$ is the velocity of the center of mass of the entire gas cloud. Notice how we subtract the center of mass velocity? This is the crucial step. We only care about how the molecules are buzzing around relative to each other, not how the whole swarm is moving together.

When the lorry moves with a uniform velocity $\vec{v}$, it imparts this exact same velocity to the container, and consequently, to every single gas molecule inside it.

Let's look at the math. The new velocity of each molecule becomes ${\vec{v}}_i^{\prime }={\vec{v}}_i+\vec{v}$.

Similarly, the entire center of mass of the gas is now moving with the lorry, so its new velocity is ${\vec{v}}_{cm}^{\prime }={\vec{v}}_{cm}+\vec{v}$.

Now, let's calculate the relative velocity that actually determines temperature:

The uniform velocity $\vec{v}$ completely cancels out!

Because the relative random velocities of the molecules remain entirely unchanged, the chaotic, disordered collisions happening inside the container are exactly the same as they would be if the lorry were at rest.

Therefore, the internal random kinetic energy is constant, and the temperature of the gas remains exactly the same.

Think of it like a swarm of bees inside a moving train. The train might be moving at $100\text{ km/hr}$, but the bees are still just buzzing around each other at their normal speed. The temperature only measures the "buzzing," not the train's speed. However, if the train were to suddenly crash, that organized forward energy would violently convert into chaotic buzzing, and then the temperature would spike!