- #1

ThEmptyTree

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- Homework Statement:
- A freight car of mass ##m_c## contains sand of mass ##m_s## . At ##t = 0## a constant horizontal force of magnitude ##F## is applied in the direction of rolling and at the same time a port in the bottom is opened to let the sand flow out at the constant rate ##b = \frac{dm_s}{dt}##. Find the speed of the freight car when all the sand is gone (Figure 12.6). Assume that the freight car is at rest at ##t = 0## .

- Relevant Equations:
- $$\overrightarrow{F}=\frac{d\overrightarrow{p}}{dt}$$

$$\overrightarrow{p_i}=\overrightarrow{0},~\overrightarrow{p_f}=m_c\overrightarrow{v_f}$$

$$\overrightarrow{p_f}-\overrightarrow{p_i}=\int\limits_{t_i}^{t_f}\overrightarrow{F}dt$$

$$t_i=0,~t_f=\frac{m_s}{b}$$

$$m_c\overrightarrow{v_f}=\overrightarrow{F}\frac{m_s}{b}\Rightarrow v_f=\frac{Fm_s}{bm_c}$$

However, their solution uses differential analysis for states ##t## and ##t+\Delta{t}##, yielding

$$v(t)=\frac{F}{b}ln\Big(\frac{m_c+m_s}{m_c+m_s-bt}\Big)$$

For ##t=t_f=\frac{m_s}{b}## according to their solution

$$v_f=\frac{F}{b}ln\Big(1+\frac{m_s}{m_c}\Big)$$

Notice that using approximation ##ln(1+x)\approx x## for small ##x## their solution becomes equivalent to mine.

What am I doing wrong? I am confused. What am I skipping when trying to analyze only the initial and final state?

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