1. The problem statement, all variables and given/known data A train wagon of mass M moves on a rail with constant velocity V (without friction). It passes a sand hopper which pours sand in the wagon at constant rate C [kg/s]. The sand falls vertically so it does not transfer any horizontal momentum to the wagon. The length of the wagon is L [m]. Question 1: Determine the velocity of the train wagon as a function of time V(t) while it is under the sand hopper. Question 2: Show that the total time T that it takes for the wagon to fully pass the hopper machine is T=M/C*(exp(CL/MV)-1) 2. Relevant equations Conservation of momentum MwagonV0=(Mwagon+msand)*V1 F=ma=dP/dt (Newton's second law) t=msand/C Straight-line motion equations vx=v0x+axt (Equation 1) x=x0+v0xt+½axt2 (Equation 2) vx2=v0x2+2a(x-x0) (Equation 3) x-x0=½(v0x+vx)t (Equation 4) 3. The attempt at a solution The first question was pretty easy. Because I need the answer from question 1 to solve question 2 I want to make sure that I have the right answer for question 1. Question 1: Using conservation of momentum I got MwagonV0=(Mwagon+msand)*V1 and the mass of the sand can be written as Ct. Subbing that in you get MwagonV0=(Mwagon+Ct)*Vt. This gives an equation for Vt -> Vt=Mwagon*V0/(Ct+Mwagon) Question 2: This one is tricky and it definitely needs integrating if you look at the equation. I have a feeling there are multiple ways to approach this but every approach I did ended up in chaotic equations that did not help. First I used Newton's second law --> F=dP/dt Pt=Mtvt with the mass being dependent of time because of Mwagon+Ct dP/dt=Mtdv/dt+vtdm/dt (dm/dt is the change in mass per unit time which is C) F=Mtdv/dt+Cvt When isolating dV/dt and integrating that I got V=(F-CV)ln(M2+C2T2) M2+C2T2=exp(V/(F-CV) This is definitely going in the wrong direction and I got no L in the equations. Then I started with equation 2 with x being L --> L=V0t+½at2. Acceleration is dV/dt which is CMV/((Ct+M)2) but subbing this in gives again nothing I can work with. Then I used equation 3 which resulted in (MV)2/((M+Ct)2)=V02+2(dV/dt)*L Isolating dV/dt gives -C2t2V2/(2(M2L+C2T2L)=dV/dt This is also a pain to integrate but I used an integration calculator and I ended up with an arctan in my equation. This is going nowhere and I would really appreciate some help.