(2) The so-called rocket equation is v – vo = Vez In(mo/m), where v and m are the speed and mass of the rocket at a certain time, vo and mo are the corresponding initial values, and vez is the exhaust speed of the burning fuel relative to the rocket. A certain two-stage rocket carries 70% of its initial mass as fuel. In the first stage, it burns 40% of its fuel. In the second stage, the rocket first ejects the first-stage fuel tank, which has mass 0.2mo, and then burns the rest of its fuel. (a) In terms of ver, what is the final speed of the rocket ? (b) Had all the fuel been burned in a single stage, without ejecting the first stage fuel tank, what would have been the final speed ?
(2) The so-called rocket equation is v – vo = Vez In(mo/m), where v and m are the speed and mass of the rocket at a certain time, vo and mo are the corresponding initial values, and vez is the exhaust speed of the burning fuel relative to the rocket. A certain two-stage rocket carries 70% of its initial mass as fuel. In the first stage, it burns 40% of its fuel. In the second stage, the rocket first ejects the first-stage fuel tank, which has mass 0.2mo, and then burns the rest of its fuel. (a) In terms of ver, what is the final speed of the rocket ? (b) Had all the fuel been burned in a single stage, without ejecting the first stage fuel tank, what would have been the final speed ?
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![**Rocket Equation and Two-Stage Rocket Problem**
The so-called rocket equation is given by:
\[ v - v_0 = v_{\text{ex}} \ln(m_0/m) \]
where \( v \) and \( m \) are the speed and mass of the rocket at a certain time, \( v_0 \) and \( m_0 \) are the corresponding initial values, and \( v_{\text{ex}} \) is the exhaust speed of the burning fuel relative to the rocket.
**Problem Description:**
A certain two-stage rocket carries 70% of its initial mass as fuel. In the first stage, it burns 40% of its fuel. In the second stage, the rocket first ejects the first-stage fuel tank, which has a mass of \( 0.2m_0 \), and then burns the rest of its fuel.
**Questions:**
(a) In terms of \( v_{\text{ex}} \), what is the final speed of the rocket?
(b) Had all the fuel been burned in a single stage, without ejecting the first-stage fuel tank, what would have been the final speed?
**Solution Approach:**
- To solve (a), calculate the remaining mass at each stage and apply the rocket equation considering the ejection of the first-stage fuel tank.
- For (b), consider the scenario where all the fuel is burned without stage separation, and calculate the final speed using the rocket equation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F001e012b-4e08-44ad-a9d9-122da7b59bd1%2F6f703580-9368-49b4-b6bd-7e83a776d26a%2Fkcwl3uf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Rocket Equation and Two-Stage Rocket Problem**
The so-called rocket equation is given by:
\[ v - v_0 = v_{\text{ex}} \ln(m_0/m) \]
where \( v \) and \( m \) are the speed and mass of the rocket at a certain time, \( v_0 \) and \( m_0 \) are the corresponding initial values, and \( v_{\text{ex}} \) is the exhaust speed of the burning fuel relative to the rocket.
**Problem Description:**
A certain two-stage rocket carries 70% of its initial mass as fuel. In the first stage, it burns 40% of its fuel. In the second stage, the rocket first ejects the first-stage fuel tank, which has a mass of \( 0.2m_0 \), and then burns the rest of its fuel.
**Questions:**
(a) In terms of \( v_{\text{ex}} \), what is the final speed of the rocket?
(b) Had all the fuel been burned in a single stage, without ejecting the first-stage fuel tank, what would have been the final speed?
**Solution Approach:**
- To solve (a), calculate the remaining mass at each stage and apply the rocket equation considering the ejection of the first-stage fuel tank.
- For (b), consider the scenario where all the fuel is burned without stage separation, and calculate the final speed using the rocket equation.
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