Question

The total mass of cart is 2 metric ton in which 1 metric ton sand is loaded. System is initially atrest. The sand leaks out of the cart (from the hole at Bottom) at the rate of 0.5 kg/s and an external horizontal force of 10N is acting on it. The final
velocity of cart when the sand completely leaks out,
is :-
(1) 13.86 m/s
(2) 14.26 m/s
(3) 20.08 m/s
(4) 16.39 m/s​

Answer 1

let after time t, loaded mass becomes , m = $m_0-\mu t$, where μ is rate of sand leaked out from the cart.

from Newton's 2nd law,

F = ma

= $(m_0-\mu t)$a

⇒a = F/$(m_0-\mu t)$

⇒dv/dt = F/$(m_0-\mu t)$

⇒∫dv = F∫dt/$(m_0-\mu t)$

⇒v = F/μ ln$\left[\frac{m_0}{(m_0-\mu t)}\right]$

now, putting,$m_0$ = 2 ma

etric ton and $(m_0-\mu t)$ = 1metric ton

and μ = 0.5 kg/s and F = 10N

so, v = 10/0.5 ln[2/1]

= 20ln2

= 20 × 0.693

= 13.86 m/s

hence, option (1) is correct choice.

Answer 2

$\huge\bold\purple{Answer:-}$

(1) 13.86 m/s