- #1
Hamiltonian
- 296
- 193
- Homework Statement
- A cart of mass ##M_0## is moving with a velocity ##v_0##. At t = 0. water starts pouring on the cart from the container above the cart at the rate ##\lambda##kg/sec. Find the velocity of the cart as a function of time.
- Relevant Equations
- -
At time t = 0, the mass of the cart is ##M_0## and velocity is ##v_0## in a time interval ##dt## let a mass of ##dm## be added to the cart due to the pouring water and let the reduction in speed be ##dv##
##\lambda = dm/dt##
applying conservation of momentum from the ground frame gives $$M_0 v_0 = (M_0 + dm)(v_0 - dv)$$
however on substituting ##dm## in terms of ##\lambda## and ##dt## gives a differential equation that I am unable to solve.
In class we were given a formula $$ F_{thrust} = v_{relative} \frac{dm}{dt}$$
we derived this formula to give the thrust of a rocket(but it can be used for any variable mass system with a few tweaks)
using this formula here gives $$\lambda (-v) = (M_0 + \lambda t)(dv/dt)$$
solving this equation yields the correct answer(##v = \frac{M_0 v_0}{M_0 + \lambda t}##). Instead of blindly applying the above-stated formula on the given problem I tried to derive it in the process of solving the problem but I am ending up with a rather weird-looking equation.
I also figured out a much simpler method that does not require any calculus its simple directly applying the conservation of momentum $$M_0 v_0 = (M_0 + \lambda t)v$$ which gives the correct answer but I want to know why the first method I am trying to use isn't working.
##\lambda = dm/dt##
applying conservation of momentum from the ground frame gives $$M_0 v_0 = (M_0 + dm)(v_0 - dv)$$
however on substituting ##dm## in terms of ##\lambda## and ##dt## gives a differential equation that I am unable to solve.

In class we were given a formula $$ F_{thrust} = v_{relative} \frac{dm}{dt}$$
we derived this formula to give the thrust of a rocket(but it can be used for any variable mass system with a few tweaks)
using this formula here gives $$\lambda (-v) = (M_0 + \lambda t)(dv/dt)$$
solving this equation yields the correct answer(##v = \frac{M_0 v_0}{M_0 + \lambda t}##). Instead of blindly applying the above-stated formula on the given problem I tried to derive it in the process of solving the problem but I am ending up with a rather weird-looking equation.
I also figured out a much simpler method that does not require any calculus its simple directly applying the conservation of momentum $$M_0 v_0 = (M_0 + \lambda t)v$$ which gives the correct answer but I want to know why the first method I am trying to use isn't working.