A gas has an initial root-mean-square velocity of 900 m/s. If the pressure of the gas increases by a factor of nine while its density remains constant, what is the new root-mean-square velocity?

Step 1: Identify the given formula and initial conditions. The given formula for pressure P in terms of density and root-mean-square velocity c is: P = (1)/(3) c^2 The initial root-mean-square velocity is c_1 = 900 m s^-1. Step 2: Analyze the changes in the gas properties. The gas is heated such that its pressure increases by a factor of nine, so P_2 = 9 P_1. The volume of the cylinder does not change, which means V_1 = V_2. Since density = (m)/(V) and the mass m of the gas remains constant, and the volume V also remains constant, the density must be constant throughout the process (_1 = _2 = ). Step 3: Set up equations for the initial and final states. For the initial state: P_1 = (1)/(3) c_1^2 For the final state: P_2 = (1)/(3) c_2^2 Step 4: Substitute the relationship between P_1 and P_2 into the equations. We know P_2 = 9 P_1. Substitute the expressions for P_1 and P_2: 9 ( (1)/(3) c_1^2 ) = (1)/(3) c_2^2 Step 5: Solve for the new root-mean-square velocity c_2. Cancel out the common terms (1)/(3) and from both sides: 9 c_1^2 = c_2^2 Take the square root of both sides: sqrt(9 c)_1^2 = sqrt(c)_2^2 3 c_1 = c_2 Now, substitute the value of c_1: c_2 = 3 × 900 m s^-1 c_2 = 2700 m s^-1 The new numerical value for the root-mean-square velocity is 2700 m s^-1. The final answer is A: 2700.