Total time wagon needs to fully pass the hopper

[freutel]

Homework Statement

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)

Homework Equations

• Conservation of momentum M_wagonV₀=(M_wagon+m_sand)*V₁

• F=ma=dP/dt (Newton's second law)

• t=m_sand/C

• Straight-line motion equations

v_x=v_0x+a_xt (Equation 1)

x=x₀+v_0xt+½a_xt² (Equation 2)

v_x²=v_0x²+2a(x-x₀) (Equation 3)

x-x₀=½(v_0x+v_x)t (Equation 4)

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 M_wagonV₀=(M_wagon+m_sand)*V₁ and the mass of the sand can be written as Ct. Subbing that in you get M_wagonV₀=(M_wagon+Ct)*V_t. This gives an equation for V_t -> V_t=M_wagon*V₀/(Ct+M_wagon)

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
P_t=M_tv_t with the mass being dependent of time because of M_wagon+Ct

dP/dt=M_tdv/dt+v_tdm/dt (dm/dt is the change in mass per unit time which is C)

F=M_tdv/dt+Cv_t

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.

 

[TSny]

Hi, freutel.

The constant acceleration equations that you wrote as relevant equations are not applicable in this problem.

You're making it more difficult than necessary. You found from question 1 that V_t=M_wagon*V₀/(Ct+M_wagon).
Can you see what to do if you write V_t = dx/dt?

 

[freutel]

I always fail to see the easiest way to solve the problem..
Ok so I did V_t=dx/dt which gives x=((M_wagonV)/C)*ln(M_wagon+Ct)
The x is of course L so I isolated everything from the natural log which gives LC/(M_wagonV)=ln(M_wagon+Ct)
This results in M_wagon+Ct=exp(CL/M_wagonV).
Now I get for t --> t=(1/C)*exp((CL/M_wagonV)-M/C.

It appears I am missing a factor M at the exponential function... I'm looking for that missing M but I cannot find it. Did I oversee something or could it be that the given equation has a factor M too many?

 

[freutel]

Hi, freutel.

The constant acceleration equations that you wrote as relevant equations are not applicable in this problem.

You're making it more difficult than necessary. You found from question 1 that V_t=M_wagon*V₀/(Ct+M_wagon).
Can you see what to do if you write V_t = dx/dt?

I know you are not supposed to give full answers but I really cannot find that missing M. Can you please help me?

 

[TSny]

When you integrate you can either do indefinite integrals or definite integrals.

If you use indefinite integrals, you will have an arbitrary constant of integration that you will need to determine.

If you use definite integrals, then make sure you use the appropriate limits of integration for x and t and evaluate the integrals at both the upper and lower limits.

 

[freutel]

Yes! I made a little mistake, I forgot to set the boundaries from 0 to T but now I got it!
Thank you very much, TSny!

 

[TSny]

OK. Good work!