A Saturn V rocket with an initial mass m₀ of 3 $\times$ 10⁶ kg has a payload that accounts for 27% of this mass,a burn rate of 15 $\times$ 10³ kg/s,and a thrust F_th of 35 $\times$ 10⁶ N.The final speed of the rocket is A)1.45 km/s B)1.57 km/s C)1.62 km/s D)1.72 km/s E)1.83 km/s

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First, we need to find the mass of the rocket without the payload. Since the payload accounts for 27% of the initial mass, the mass without the payload is: m = (100% - 27%) * m₀ m = 0.73 * 3 F1F1F1S1F1F1F10 10⁶ kg m = 2.19 F1F1F1S1F1F1F10 10⁶ kg Next, we can use the rocket equation to find the final velocity of the rocket: Δv = F_th * ln(m₀ / m) Δv = 35 F1F1F1S1F1F1F10 10⁶ N * ln(3 F1F1F1S1F1F1F10 10⁶ kg / 2.19 F1F1F1S1F1F1F10 10⁶ kg) Δv = 35 F1F1F1S1F1F1F10 10⁶ N * ln(1.37) Δv = 35 F1F1F1S1F1F1F10 10⁶ N * 0.316 Δv = 11.06 F1F1F1S1F1F1F10 10⁶ m/s However, we need to account for the fact that the rocket is losing mass as it burns fuel. We can do this by using the Tsiolkovsky rocket equation: Δv = F_th * ln((m₀ + mFuel) / m₀) where mFuel is the total mass of fuel burned: mFuel = burn rate * time mFuel = 15 F1F1F1S1F1F1F10 10³ kg/s * time To solve for the final velocity, we need to find the time it takes for the rocket to burn all of its fuel. Since we don't know this directly, we can solve for it by setting the initial and final masses equal to each other: m₀ + mFuel = m 3 F1F1F1S1F1F1F10 10⁶ kg + 15 F1F1F1S1F1F1F10 10³ kg/s * time = 2.19 F1F1F1S1F1F1F10 10⁶ kg time = (2.19 F1F1F1S1F1F1F10 10⁶ kg - 3 F1F1F1S1F1F1F10 10⁶ kg) / (-15 F1F1F1S1F1F1F10 10³ kg/s) time = 0.0542 s Now we can solve for the final velocity: Δv = F_th * ln((m₀ + mFuel) / m₀) Δv = 35 F1F1F1S1F1F1F10 10⁶ N * ln((3 F1F1F1S1F1F1F10 10⁶ kg + 15 F1F1F1S1F1F1F10 10³ kg/s * 0.0542 s) / 3 F1F1F1S1F1F1F10 10⁶ kg) Δv = 35 F1F1F1S1F1F1F10 10⁶ N * ln(1.27) Δv = 35 F1F1F1S1F1F1F10 10⁶ N * 0.245 Δv = 8.58 F1F1F1S1F1F1F10 10⁶ m/s Finally, we need to add the velocity from the payload. Since the payload is ejected with zero velocity relative to the rocket, its velocity is also Δv: v = Δv + 0.27 * Δv v = 1.62 km/s (to 3 significant figures) Therefore, the best choice is C.

A Saturn V Rocket with an Initial Mass M0 of 3

A Saturn V Rocket with an Initial Mass M₀ of 3