A rocket with initial mass of 3.0 × 10⁴ kg fires its engines at t = 0.The exhaust speed of the rocket is 3.9 × 10³ m/s.If the velocity of the rocket at t = 200 s is 1 500 m/s,how much fuel did it consume? A)9.6 × 10³ kg B)3.0 × 10⁴ kg C)4.4 × 10⁴ kg D)4.8 × 10³ kg

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Quizplus · Accepted Answer

To find the mass of the fuel consumed, we can use the rocket equation, which is derived from the conservation of momentum and is given by the Tsiolkovsky rocket equation:$$ \Delta v = v_e \ln \left( \frac{m_i}{m_f} \right) $$where:- $\Delta v$ is the change in velocity of the rocket (1500 m/s),- $v_e$ is the exhaust velocity (3.9 × 10^3 m/s),- $m_i$ is the initial mass of the rocket (3.0 × 10^4 kg),- $m_f$ is the final mass of the rocket.Rearranging the equation to solve for $m_f$:$$ m_f = m_i \times e^{-\frac{\Delta v}{v_e}} $$Plugging in the given values:$$ m_f = 3.0 \times 10^4 \times e^{-\frac{1500}{3.9 \times 10^3}} $$$$ m_f = 3.0 \times 10^4 \times e^{-0.3846} $$$$ m_f \approx 3.0 \times 10^4 \times 0.6806 $$$$ m_f \approx 2.0418 \times 10^4 \, \text{kg} $$The mass of the fuel consumed is the initial mass minus the final mass:$$ \text{Fuel consumed} = m_i - m_f $$$$ \text{Fuel consumed} = 3.0 \times 10^4 \, \text{kg} - 2.0418 \times 10^4 \, \text{kg} $$$$ \text{Fuel consumed} \approx 9.6 \times 10^3 \, \text{kg} $$Therefore, the correct answer is 9.6 × 10^3 kg, which corresponds to choice A.

A Rocket with Initial Mass of 3