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Question

The velocity of sound in a gas is measured to be 340 m/s. If the density of the gas is doubled while the pressure remains constant, what will be the new velocity of sound?

SUBHANI · Accepted Answer

## CORE INFORMATION
✓ **Answer**: A
## CONCEPT FRAMEWORK
	**Primary Concept**: Speed of sound in a gas
	**Related Topics**: Wave propagation, Gas laws, Thermodynamics
	**Prerequisites**: Understanding of the relationship between speed of sound, pressure, and density of a gas.
## SOLUTION PATHWAY
**Given Information**:
	Initial velocity of sound ($v_1$) = 340 m/s
	Initial density ($ho_1$) = $ho$
	Final density ($ho_2$) = $2ho$
	Pressure remains constant ($P_1 = P_2$)
**Method & Steps**:
	1. **Recall the formula for the speed of sound in a gas:**
		Reasoning: The speed of sound in a gas is given by $v = \sqrt{\frac{\gamma P}{ho}}$, where $\gamma$ is the adiabatic index (ratio of specific heats), $P$ is the pressure, and $ho$ is the density.
		Formula/Rule: $v = \sqrt{\frac{\gamma P}{ho}}$
	2. **Set up a ratio of initial and final velocities:**
		Working: Since the pressure remains constant, we can write:
		$\frac{v_2}{v_1} = \sqrt{\frac{ho_1}{ho_2}}$
		Key Point: The adiabatic index ($\gamma$) and pressure ($P$) cancel out because they are constant.
	3. **Calculate the new velocity:**
		Working: Substitute the given values into the ratio:
		$\frac{v_2}{340} = \sqrt{\frac{ho}{2ho}} = \sqrt{\frac{1}{2}}$
		$v_2 = 340 	imes \sqrt{\frac{1}{2}} = 340 	imes \frac{1}{\sqrt{2}} \approx 340 	imes 0.707 \approx 240.4$ m/s
		Result: $v_2 \approx 240.4$ m/s
		Verification: Since the density has increased while the pressure remains constant, the speed of sound should decrease. The calculated value is less than the initial velocity, which is consistent with this expectation.
## OPTION ANALYSIS
[A] ✓: 240.4 m/s - This is the correct answer, as calculated above.
[B] :x: 480.8 m/s - This is incorrect.  The speed of sound decreases when density increases at constant pressure.
[C] :x: 170 m/s - This is incorrect.  This value is too low.
[D] :x: 340 m/s - This is incorrect. The speed of sound changes when density changes at constant pressure.
## LEARNING AIDS
	**Common Mistakes**:
		1. Incorrectly assuming that the speed of sound is directly proportional to density: Remember that the speed of sound is inversely proportional to the square root of density.
		2. Forgetting to consider the square root in the formula: Ensure to take the square root of the density ratio when calculating the new velocity.
	**Quick Tips**:
		1. Remember the relationship:  Higher density leads to lower speed of sound at constant pressure.
		2. Visualize the formula:  $v \propto \frac{1}{\sqrt{ho}}$

SUBHANI · Answer

The correct Option is "240.4 m/s"

SUBHANI · Answer

480.8 m/s

SUBHANI · Answer

170 m/s

SUBHANI · Answer

340 m/s

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