Question

A molecule of mass $m$ moving with velocity $v$ makes 5 elastic collisions with a wall of container per second. The change in momentum of the wall per second in 5 collisions will be

• A. $10\ mv$
• B. $5\ mv$
• C. $15\ mv$
• D. $\frac{1}{10}\ mv$

Hint:

Always remember that momentum is a vector quantity! A common trap is calculating the change in speed as $v - v = 0$, or forgetting the factor of 2 and choosing $5\ mv$ (option B). Because the direction reverses, the magnitude of the change per collision is always doubled ($2mv$).

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The problem tracks a gas molecule colliding periodically against a flat container boundary. We are given the molecule's mass ($m$), its speed ($v$), and the collision rate ($f = 5\text{ collisions/second}$). We need to find the total momentum transferred to the wall per second.

Step 2: Key Formula or Approach:
For a perfectly elastic collision against a rigid wall, the molecule rebounds with the exact same speed but in the opposite direction.
The change in momentum of a single molecule ($\Delta p_{\text{molecule}}$) along the normal axis is: $$\Delta p_{\text{molecule}} = p_{\text{final}} - p_{\text{initial}} = (-mv) - (mv) = -2mv$$ By Newton's Third Law (Action-Reaction), the momentum transferred to the wall per collision is equal in magnitude and opposite in sign: $$\Delta p_{\text{wall}} = +2mv$$ The total change in momentum per second is the momentum from one collision multiplied by the number of collisions per second.

Step 3: Detailed Explanation:
Calculate the momentum change for the wall during a single collision event: $$\Delta p_{\text{single}} = 2mv$$ Since the molecule undergoes exactly 5 independent collisions every second, multiply the single-collision transfer by 5: $$\Delta p_{\text{total}} = 5 \times \Delta p_{\text{single}}$$ $$\Delta p_{\text{total}} = 5 \times (2mv) = 10\ mv$$ This matches the value in option (A).

Step 4: Final Answer:
The total momentum transferred to the wall per second is $10\ mv$, which corresponds to option (A).