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Standard IX

Physics

Buoyant Force

A cart loaded...

Question

A cart loaded with sand moves along a horizontal plane due to a constant force $F$ in the direction of velocity. Through a small hole in the bottom, sand spills out at a rate of $m k g s^{- 1}$ . Find the velocity of the cart at any time $t$ . Initial mass of the cart is $m_{0}$ .

\frac{F}{m} l o g_{e} \frac{m_{0}}{m_{0} - m t}

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\frac{F}{m_{0}} l o g_{e} \frac{m_{0}}{m_{0} - m t}

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\frac{F}{m_{0}} t

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\frac{F}{m} t

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Solution

The correct option is A $\frac{F}{m} l o g_{e} \frac{m_{0}}{m_{0} - m t}$
Mass of the cart at instant $t = (m_{0} - m t)$
now, $(m_{0} - m t) a = F$ and $a = \frac{d v}{d t}$
$\frac{d v}{d t} = \frac{F}{m_{0} - m t}$
or, $v = \int_{0}^{t} \frac{F}{m_{0} - m t} d t = - \frac{F}{m} [l o g_{e} (m_{0} - m t) - l o g_{e} (m_{0})]$
or, $v = - \frac{F}{m} l o g_{e} \frac{(m_{0} - m t)}{m_{0}} = \frac{F}{m} l o g_{e} \frac{m_{0}}{(m_{0} - m t)}$

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A cart loaded with sand moves along a horizontal plane due to a constant force F in the direction of velocity. Through a small hole in the bottom, sand spills out at a rate of m kgs−1. Find the velocity of the cart at any time t. Initial mass of the cart is m0.

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A cart loaded with sand moves along a horizontal plane due to a constant force $F$ in the direction of velocity. Through a small hole in the bottom, sand spills out at a rate of $m k g s^{- 1}$ . Find the velocity of the cart at any time $t$ . Initial mass of the cart is $m_{0}$ .