How to Calculate Impulse in a Momentum Problem?

[Ariano AnnaG]

Homework Statement

HALLIDAY CHAPTER 9, momentum and impulse

Statement of the problem:
Two vessels are proceeding in the same direction through a river. Their velocities are respectively
10 km/h and 20 km/h. While the second boat is getting over the first one, a men is throwing 1000kg of charcoal per minute from the slower the faster.

Which variation of the force has to be applied to the motor of the slower boat and which to the motor of the faster, in order to keep their velocities constant? This, considering that the transfer of the charcoal doesn't interfere with the motion of the vessels and that the friction force doesn’t depend on the charge of the boats.

Solution given by the book: (a) 46N (b) none

DATAS:
V1i= 10km/h = 10/3,6 m/s velocity of the first boat
V2i= 20km/h =20/3,6 m/s velocity of the second boat
k= 1000kg/min = 1000kg/60s quantity of charcoal moved by the men in a minute

Relevant Equations

impulse-momentum theorem general equation: FΔt=Δq

1)In the first case the mass changes, in this manner:

m1= m1i - k Δt

impulse-momentum theorem:

F1Δt=Δq

FΔt=m1V1-m1iV1i

I now can replace m1 with its definition and obtain:
F1Δt= (m1i - k Δt)V1 -m1iV1i = m1iV1i - kΔtV1- m1iV1i = - kΔtV1

It is possible to remove Δt as it appears in both members of the equation:
a) F1= -kV1= -1000kg/60s x 10/3,6 m/s= -46 N2)In the second case the mass changes, in this manner:

m2= m2i +k Δt

impulse-momentum theorem:

F2Δt=Δq

F2Δt=m2V2 -m2iV2i

I can now replace m1 with its definition and obtain:
F2Δt= (m2i + k Δt) V2 - m2iV2i = m2iV2i + kΔtV2- m2iV2i = +kΔtV2

It is possible to remove Δt as it appears in both members of the equation:
b) F2= kV2= +1000kg/60s x 20/3,6 m/s= +92,6 N ≅ 93 N

I thought I had to use the impulse-momentum theorem in the second case, like I did before to solve the first request of the problem, but my result (93 N) has nothing to do with the solution (b) of the book, (0 N, there isn’t a force).

What am I not considering or forgetting? Thank you in advance for your disposability and patience

 

[haruspex]

FΔt=m1V1-m1iV1i

Conservation applies to a closed system. Since mass does not change in a closed system there cannot be two different masses in the above equation.
You need to consider the momentum of the coal being removed.

However, the problem is missing information. Is the coal transferred perpendicularly to to the boats' motion or at some angle? If perpendicularly then the answers are swapped. The slower boat needs no thrust.

 

[Ariano AnnaG]

Thank you haruspex for your reply! So it would be FΔt=m1V1-m1V1i ? I'd say the coal is transferred perpendicularly, because there isn't a given angle in the problem statement. Is the book swapping solution (a) with solution (b) or is it me, dealing with the procedure?

 

[haruspex]

Thank you haruspex for your reply! So it would be FΔt=m1V1-m1V1i ? I'd say the coal is transferred perpendicularly, because there isn't a given angle in the problem statement. Is the book swapping solution (a) with solution (b) or is it me, dealing with the procedure?

Consider the system consisting of a shovelful of coal about to be transferred, the remaining coal on the slower vessel, and the slower vessel itself.
If it is being transferred perpendicularly, what impulse is there on the subsystem consisting of the last two of those?

 

[jbriggs444]

Conservation applies to a closed system. Since mass does not change in a closed system there cannot be two different masses in the above equation.
You need to consider the momentum of the coal being removed.

However, the problem is missing information. Is the coal transferred perpendicularly to to the boats' motion or at some angle? If perpendicularly then the answers are swapped. The slower boat needs no thrust.

Perpendicularly according to what frame of reference? Slow boat, fast boat, river, other?

Workmen, being workmen will go for the easiest approach. They will normally toss perpendicular to the tossing boat -- the slower boat in this case. The sacks or scooploads of coal will be seen to arrive from a forward angle as judged by the receiving boat because it is faster.

 

[haruspex]

Perpendicularly according to what frame of reference?

Fair question. I meant relative to the boat from which the coal is being transferred.

 

[Ariano AnnaG]

Consider the system consisting of a shovelful of coal about to be transferred, the remaining coal on the slower vessel, and the slower vessel itself.
If it is being transferred perpendicularly, what impulse is there on the subsystem consisting of the last two of those?

Thank you, you were really helpful, with your advice I was finally able to solve it!