Question

Consider a gas enclosed in a box. A molecule of mass $m$, having a velocity $-5\hat{i}+3\hat{j}+4\hat{k}$ collides with a wall parallel to the $xz$ plane. What will be its velocity after collision and its change in momentum? If $\hat{i}, \hat{j} \& \hat{k}$ are unit vectors along the $x, y \& z$ axis.

• A. -5î+3ĵ+4k̂
• B. 5î+3ĵ+4k̂
• C. 5î-3ĵ+4k̂
• D. 5î+3ĵ-4k̂

Answer 1

Answer: (C)

5î-3ĵ+4k̂

Answer 2

The problem describes an elastic collision of a gas molecule with a wall parallel to the $xz$ plane. In an elastic collision with a stationary wall, the component of the molecule's velocity perpendicular to the wall reverses direction, while the components parallel to the wall remain unchanged.

Given:

• Mass of the molecule = $m$
• Initial velocity of the molecule, $\vec{v}_i = -5\hat{i} + 3\hat{j} + 4\hat{k}$
• The wall is parallel to the $xz$ plane. This means the normal to the wall is along the y-axis.

1. Determine the velocity after collision ($\vec{v}_f$):

The initial velocity components are:

• x-component: $-5\hat{i}$ (parallel to the $xz$ plane)
• y-component: $3\hat{j}$ (perpendicular to the $xz$ plane)
• z-component: $4\hat{k}$ (parallel to the $xz$ plane)

According to the rules of elastic collision with a wall:

• The components parallel to the wall (x and z components) remain unchanged. So, the x-component remains $-5\hat{i}$ and the z-component remains $4\hat{k}$.
• The component perpendicular to the wall (y-component) reverses its direction. So, $3\hat{j}$ becomes $-3\hat{j}$.

Therefore, the velocity after collision, $\vec{v}_f$, will be:

$$\vec{v}_f = -5\hat{i} - 3\hat{j} + 4\hat{k}$$

2. Determine the change in momentum ($\Delta \vec{p}$):

The change in momentum is given by the difference between the final momentum and the initial momentum:

$$\Delta \vec{p} = \vec{p}_f - \vec{p}_i = m\vec{v}_f - m\vec{v}_i = m(\vec{v}_f - \vec{v}_i)$$

Substitute the initial and final velocities:

$$\Delta \vec{p} = m[(-5\hat{i} - 3\hat{j} + 4\hat{k}) - (-5\hat{i} + 3\hat{j} + 4\hat{k})]$$ $$\Delta \vec{p} = m[-5\hat{i} - 3\hat{j} + 4\hat{k} + 5\hat{i} - 3\hat{j} - 4\hat{k}]$$

Combine the components:

$$\Delta \vec{p} = m[(-5+5)\hat{i} + (-3-3)\hat{j} + (4-4)\hat{k}]$$ $$\Delta \vec{p} = m[0\hat{i} - 6\hat{j} + 0\hat{k}]$$ $$\Delta \vec{p} = -6m\hat{j}$$

Conclusion:

The velocity after collision is $\vec{v}_f = -5\hat{i} - 3\hat{j} + 4\hat{k}$.

The change in momentum is $\Delta \vec{p} = -6m\hat{j}$.

Upon reviewing the provided options for the velocity after collision:

• $-5\hat{i}+3\hat{j}+4\hat{k}$ (This is the initial velocity)
• $5\hat{i}+3\hat{j}+4\hat{k}$
• $5\hat{i}-3\hat{j}+4\hat{k}$
• $5\hat{i}+3\hat{j}-4\hat{k}$

None of the given options match the calculated final velocity $\vec{v}_f = -5\hat{i} - 3\hat{j} + 4\hat{k}$.

There appears to be an error in the question's options. If we assume a common typo where the initial x-component of velocity was intended to be positive ($5\hat{i}$) instead of negative ($-5\hat{i}$), then the calculation would lead to one of the options.

If $\vec{v}_i = 5\hat{i} + 3\hat{j} + 4\hat{k}$, then $\vec{v}_f = 5\hat{i} - 3\hat{j} + 4\hat{k}$, which is the third option.

However, based on the question as written, there is no correct option. Since I must choose one, and assuming the most probable intended question, I will proceed with the scenario where the initial x-component was positive.

Assuming a typo in the question where initial velocity was $5\hat{i}+3\hat{j}+4\hat{k}$:

Initial velocity, $\vec{v}_i = 5\hat{i} + 3\hat{j} + 4\hat{k}$.

Final velocity, $\vec{v}_f = 5\hat{i} - 3\hat{j} + 4\hat{k}$.

This matches the third option.