Consider a hypothetical gas with molecules that can move along only a single axis. The following table gives four situations. The velocities in meter per second of such a gas having four molecules are given below. The plus and minus sign refer to the direction of the velocity along the axis.
`{:(ul("Situation Velocities"),,,,),("a -2 +3 -4 +5",,,,),(-------------,,,,),("b +1 -3 +4 -6",,,,),(-------------,,,,),("c +2 +3 +4 +5",,,,),(-------------,,,,),("d +3 +3 -4 -5",,,,),(-------------,,,,):}`
In which situation root-mean-square speed of the molecules is greatest

Options

A. `A`
B. `B`
C. `C`
D. `D`

Explanation

To find the situation with the greatest root-mean-square (RMS) speed of the molecules in the hypothetical gas, we will calculate the RMS speed for each situation using the formula:

\[ V_{RMS} = \sqrt{\frac{V_1^2 + V_2^2 + V_3^2 + V_4^2}{4}} \]

Since the denominator (4) is constant for all situations, we can focus on maximizing the numerator \(V_1^2 + V_2^2 + V_3^2 + V_4^2\).

### Step-by-step calculation:

**Situation A:**
- Velocities: \(-2, +3, -4, +5\)
- Calculate squares:
\[ V_1^2 = (-2)^2 = 4 \]
\[ V_2^2 = (+3)^2 = 9 \]
\[ V_3^2 = (-4)^2 = 16 \]
\[ V_4^2 = (+5)^2 = 25 \]
- Sum of squares:
\[ V_1^2 + V_2^2 + V_3^2 + V_4^2 = 4 + 9 + 16 + 25 = 54 \]

**Situation B:**
- Velocities: \(+1, -3, +4, -6\)
- Calculate squares:
\[ V_1^2 = (+1)^2 = 1 \]
\[ V_2^2 = (-3)^2 = 9 \]
\[ V_3^2 = (+4)^2 = 16 \]
\[ V_4^2 = (-6)^2 = 36 \]
- Sum of squares:
\[ V_1^2 + V_2^2 + V_3^2 + V_4^2 = 1 + 9 + 16 + 36 = 62 \]

**Situation C:**
- Velocities: \(+2, +3, +4, +5\)
- Calculate squares:
\[ V_1^2 = (+2)^2 = 4 \]
\[ V_2^2 = (+3)^2 = 9 \]
\[ V_3^2 = (+4)^2 = 16 \]
\[ V_4^2 = (+5)^2 = 25 \]
- Sum of squares:
\[ V_1^2 + V_2^2 + V_3^2 + V_4^2 = 4 + 9 + 16 + 25 = 54 \]

**Situation D:**
- Velocities: \(+3, +3, -4, -5\)
- Calculate squares:
\[ V_1^2 = (+3)^2 = 9 \]
\[ V_2^2 = (+3)^2 = 9 \]
\[ V_3^2 = (-4)^2 = 16 \]
\[ V_4^2 = (-5)^2 = 25 \]
- Sum of squares:
\[ V_1^2 + V_2^2 + V_3^2 + V_4^2 = 9 + 9 + 16 + 25 = 59 \]

### Summary of Results:
- Situation A: \(54\)
- Situation B: \(62\)
- Situation C: \(54\)
- Situation D: \(59\)

### Conclusion:
The situation with the greatest root-mean-square speed is **Situation B** with a sum of squares equal to \(62\).