Question

A wagon filled with sand has a hole so that sand leaks through the bottom at a constant rate $\lambda$. An external force $\overrightarrow{F}$ acts on the wagon in the direction of motion. Assuming instantaneous velocity of the wagon to be $\overrightarrow{v}$ and initial mass of system to be $m_0$, the force equation governing the motion of the wagon is

• A. $\overrightarrow{F}=m_0\frac{d\overrightarrow{v}}{dt}+\lambda \overrightarrow{v}$
• B. $\overrightarrow{F}=m_0\frac{d\overrightarrow{v}}{dt}-\lambda \overrightarrow{v}$
• C. $\overrightarrow{F}=(m_0-\lambda t)\frac{d\overrightarrow{v}}{dt}$
• D. $\overrightarrow{F}=(m_0-\lambda t)\frac{d\overrightarrow{v}}{dt}+\lambda \overrightarrow{v}$

Answer 1

The correct option is C $\overrightarrow{F}=(m_0-\lambda t)\frac{d\overrightarrow{v}}{dt}$

Suppose a wagon wheel together with it's sand has a mass $m_0$ at t=0.

Let the sand be ejected at constant rate $\lambda$=$\frac{dm}{dt}$.Also suppose the sand is ejected at constant velocity u w.r.t wagon.
At time t, the mass of wagon together with remaining sand is
M=$m_0$-$\lambda$t.
If the velocity of wagon at time t is $v$ ,the linear momentum of this mass M is
P=Mv=($m_0$-$\lambda t$)$v$
or $\overrightarrow{F}$=($m_0$-$\lambda t$)$\frac{d\overrightarrow{v}}{dt}$