Consider a cart of mass M on a frictionless surface that can hold a full tank of water with mass M.A fire-hose sprays water with a constant ejection speed `V_(w)` at a constant mass rate `r=(dm)/(dt)` and at an angle `theta` relative to the horizontal.

Options

A. The acceleration at any time t of the cart while it is spraying water is given by `(V_(w)rcostheta)/(M-rt)`
B. The speed of the cart as a function of time `(t lt M/r)is (V_(w)rt costheta)/(M-rt)`
C. The speed of the cart as a function of time `(t lt M/r)is V_(w)costheta l n(M-rt)`
D. The external horizontal force that must be applied to keep the cart stationary while spraying water is `rV_(w)costheta`

Answer

A, D

Explanation

`F_(h)=V_(w)(dm)/(dt)costheta=V_(w)rcostheta`
`a_(h)=(V_(w)rcostheta)/(M-rt)`
`(dv)/(dt)=(V_(w)rcostheta)/(M-rt)`
`underset0oversetvint dv=V_(w)rcostheta underset0overset tint(dt)/(M-rt)`
`v(t)=V_(w)rcostheta l n((M-rt)/(-r))|_(0)^(t)Rightarrow v(t)=V_(w)costheta l n(M/(M-rt))`