Question

A truck is moving with an acceleration of $2{\text{m/s}}^2$ due to a constant horizontal force $F$. Suddenly, a hail storm starts with a constant rate of mass $\mu =20\text{kg/s}$ getting deposited on the truck. Find the force needed to maintain the acceleration of the truck after $2$ seconds. Take $g=10{\text{m/s}}^2.$ Speed of the hailstorm is $2\text{m/s}$ (perpendicular to the velocity of the truck) and mass of the truck is initially $m_0=50\text{kg}$.

• A. $324\text{N}$
• B. $0\text{N}$
• C. $50\text{N}$
• D. $300\text{N}$

Answer 1

The correct option is A $324\text{N}$



Net force on the truck after time ($t$)
$ma=F-F_t$
$F_t=v_r\frac{dm}{dt}=-v\frac{dm}{dt}$
(because horizontal component of velocity of hailstorm is initially zero)

$\Rightarrow m\frac{dv}{dt}=F-v\frac{dm}{dt}$
$\Rightarrow mdv+vdm=Fdt$
$\Rightarrow d\left(mv\right)=Fdt$
After intergrating, $v=\frac{Ft}{m}$
But $m=m_0+\mu t$
$v=\frac{Ft}{m_0+\mu t}$

As we know, $a=\frac{dv}{dt}$
(differentiate both side w.r.t $t$ )
$a=\frac{m_{0}F}{(m_0+\mu t{)}^2}$
$\Rightarrow F=\frac{a(m_0+\mu t{)}^2}{m_0}$
At $t=2\text{s}$, $F=\frac{2(50+20\times 2{)}^2}{50}$
$\therefore F=324\text{N}$