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# When the mass m of a body is changing with time, Newton’s

**Chapter 1, Problem 22E**

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**QUESTION:**

When the mass m of a body is changing with time, Newton’s second law of motion becomes

\(F=\frac{d}{d t}(m v)\),

where F is the net force acting on the body and mv is its momentum. Use (17) in Problems 21 and 22.

In Problem 21, the mass m(t) is the sum of three different masses: \(m(t)=m_{p}+m_{v}+m_{f}(t)\) where \(m_{p}\) is the constant mass of the payload, \(m_{v}\) is the constant mass of the vehicle, and \(m_{f}(t)\) is the variable amount of fuel.

(a) Show that the rate at which the total mass m(t) of the rocket changes is the same as the rate at which the mass \(m_{f}(t)\) of the fuel changes.

(b) If the rocket consumes its fuel at a constant rate \(\lambda\), find m(t). Then rewrite the differential equation in Problem 21 in terms of \(\lambda\) and the initial total mass \(m(0)=m_{0}\).

(c) Under the assumption in part (b), show that the burnout time \(t_{b}>0\) of the rocket, or the time at which all the fuel is consumed, is \(t_{b}=m_{f}(0) / \lambda\), where \(m_{f}(0)\) is the initial mass of the fuel.

Text Transcription:

F=d/dt (mv)

m(t)=m_p + m_v + m_f(t)

m_p

m_v

m_f(t)

lambda

m(0)=m_0

t_b>0

t_b=m_f(0)/lambda

m_f(0)

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### Questions & Answers

**QUESTION:**

When the mass m of a body is changing with time, Newton’s second law of motion becomes

\(F=\frac{d}{d t}(m v)\),

where F is the net force acting on the body and mv is its momentum. Use (17) in Problems 21 and 22.

In Problem 21, the mass m(t) is the sum of three different masses: \(m(t)=m_{p}+m_{v}+m_{f}(t)\) where \(m_{p}\) is the constant mass of the payload, \(m_{v}\) is the constant mass of the vehicle, and \(m_{f}(t)\) is the variable amount of fuel.

(a) Show that the rate at which the total mass m(t) of the rocket changes is the same as the rate at which the mass \(m_{f}(t)\) of the fuel changes.

(b) If the rocket consumes its fuel at a constant rate \(\lambda\), find m(t). Then rewrite the differential equation in Problem 21 in terms of \(\lambda\) and the initial total mass \(m(0)=m_{0}\).

(c) Under the assumption in part (b), show that the burnout time \(t_{b}>0\) of the rocket, or the time at which all the fuel is consumed, is \(t_{b}=m_{f}(0) / \lambda\), where \(m_{f}(0)\) is the initial mass of the fuel.

Text Transcription:

F=d/dt (mv)

m(t)=m_p + m_v + m_f(t)

m_p

m_v

m_f(t)

lambda

m(0)=m_0

t_b>0

t_b=m_f(0)/lambda

m_f(0)

**ANSWER:**

Step 1 of 4

In this problem, we have to show that the total mass of the rocket changes is the same as the rate at which the mass of the fuel changes.