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```        a cart moves on a smooth horizontal surface due to an external constant force  F  the intial mass of the  the cart is M and the velocity is 0. sand falls on the cart with negligible velocity at constant rate U kg/s and sticks to the cart . find the velosity of the cart at the time T . ALSO FIND THE RATE OF INCREASE OF THE KINETIC ENERGY  OF THE CART.
```
10 years ago

```							at any time ,
a = F/(M+m),
where m = Ut,   [dm/dt = U]
now writing acceleration as dv/dt
dv/dt = F/(M+Ut)
now integrating both sides ,we get
V = F*ln(1+Ut/M)/U ....here initial conditions are at t= 0,V=0, m = 0

now k.E , k = 0.5*(M+m)V2
rate of change of kE is dk/dt
dk/dt = 0.5*U*V2 + 0.5*(M+m)*2*V*dv/dt

dk/dt = 0.5*U*V2 + (M+m)*V*dv/dt

now just substituting the values of V , m, dV/dt

we can get the rate of change of kE

```
10 years ago
```							hi,
mass of system at any time t= M+Ut
as we know that
mdv/dt =Fext. + Vrel.(of variable mass w.r.t.cart)dm/dt
so, we can write
(M + Ut)dv/dt = F + 0*U
dv = int[dt/(M/F+Ut/F] limits 0 to t
so, v = ln[(M/F+Ut/F)/(M/F)]
or v = ln[(M+Ut)/M]
hence we can find K.E. & its rate of change

```
10 years ago
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