- #1

PhysicsRock

- 37

- 4

- Homework Statement:
- A cart with mass ##m_0 = 10 \, \text{t}## is filled with ##15 \, \text{t}## of sand. It is then accelerated with a Force of ##F = 5000 \, \text{N}##. However, a hatch was left open and sand is leaking from the cart at a constant rate of ##\Phi = 0.5 \, \frac{\text{t}}{\text{s}}##. Give the speed of the cart after the sand is all gone as a function of ##F## and ##\Phi##. You may neglect friction in this problem.

- Relevant Equations:
- /

My approach is to use the definition of the Force with ##\displaystyle F = \frac{dp}{dt} = \dot{m} v + m \dot{v}##. Since ##m(t)## decreases linearly, I should be able to set ##m(t) = M - \Phi t##, thus ##F = - \Phi v + (M - \Phi t) \dot{v}##, which gives ##\displaystyle v = -\frac{ F - (M - \Phi t) \dot{v} }{\Phi}##. Here's the problem. I don't know how I am supposed to use this, as ##v## is unknown and thus ##\dot{v}## is unknown too. I tried solving this like an ODE, however, with that I got speeds of ##15,000 \frac{\text{m}}{\text{s}}##, which is obviously quite unrealistic. Anything I did wrong in the approach itself? Or is my ODE solution wrong? I calculated it to be ##\displaystyle v(t) = \frac{ F t }{M - \Phi t}##.

Thank you in advance.

Thank you in advance.