Mistake on official answer (conservation of linear momentum)

Hernaner28

1. The problem statement, all variables and given/known data

The problem says:

A mine provides a transport ystem constiting into carriages (of neglect mass) hanging on a riel free of friction.
In a specific moment two people of mass m=45kg are travelling when one of them says his breaks are not working. In that moment he was at V1=10m/s whilst the carriage ahead was at V2=8.0m/s.
Fortunately who travells in the second carriage can take a mass of 25Kg of sand and throw a fraction of it with a velocity V0=6.0m/s (respect to him).
Calculate the MINIMUM mass that the man will have to throw to avoid the colission.

** The "correct" answer is 23Kg

2. Relevant equations



3. The attempt at a solution

Well, obviously this is a conservation of linear momentum problem but I think it's not 23Kg. Here's what I did:

[tex] \displaystyle {{v}_{mass}}={{v}_{0}}+{{v}_{back}}[/tex]

That's the speed of the mass respect to the Earth. Vback is the additional speed the man should have when throwing the mass. If it asks the minimum then the man has to have a final speed of 10m/s, and therefore the "backward" speed should be 2m/s.(8+2=10):

[tex] \displaystyle {{v}_{mass}}=-6.0+2.0=-4.0[/tex]

[tex] \displaystyle {{p}_{i}}={{v}_{2i}}\left( m+{{m}_{x}} \right)[/tex]
[tex] \displaystyle {{p}_{f}}=\left( m+{{m}_{x}} \right){{v}_{2f}}+{{m}_{x}}\left( {{v}_{mass}} \right)[/tex]

m is 45Kg (man mass) and mx is the unknown thrown mass.
Notice that my initial mass is the unknown thrown mass plus the mass of the man, but solving for those values I get another answer BUT if I consider the initial mass as only the mass of the man I get 22.5 which is the correct answer. But should I consider both mass?

Thanks!

Attached Thumbnails

 

denjay

Rereading this part of the problem might help.

If you have any more questions, ask!

azizlwl

If second man drops a sand, what is the velocity of the sand with respect to him and what is velocity relative to ground ?

The final mass should be less than orginal mass since part has been thrown away.

Hernaner28

Sorry, it is a mminus sign, I typed it wrongly but I don't get the answer anyway. I get 23 Kg when I do what I told. I already know how to solve this problem, I just need you tell me what is correct, consider the whole man and sand or only the mass of the man".

Thanks!

azizlwl

You have to consider whole man and sand.
Your v_mass is wrong.

Hernaner28

Why? The man throws the mass and he sees it at 6.0 m/s, so the Vmass respect to the ground is -6.0+ Velocity that the man gains which should be 2.0m/s so that he equals the velocity of the man without breaks and that would be the minimum.

azizlwl

Take example for 1 seconds after they depart.

The man moves 10m to the right
Now relative to the man the sand moved 6m from him.

What is the distance moved according to a man on the ground?

Hernaner28

The man moved 10m to the right but saw from the ground. He actually has to gain a speed of 2m/s. Could you finally explain it? because I'm not getting it.

Thank you!

azizlwl

So we are required to have minimum 10m/s to avoid collision.
10m/s is ground speed.
6m/s is relative to the man.
Every second the man moved 10m.
To the man, the sand moved 6m

Can you find what the distance travelled by the sand relative to ground in one second?

Hernaner28

16m/s ?

azizlwl

So what is the sand's ground velocity?
Wrong for 16m/s.

Hernaner28

4 m/s ? I don't know! Please tell me... if you explain it I will understand it.

azizlwl

For 1 sec.

.------------------------ Man
0------------------------10m
|........|....................|
|........|<......6m.....>|
-------- Sand

You got the value and subtitute in your equation with man and sand.

Hernaner28

Seriously, I am not understanding those relative velocities, I need you to explain it to me.

If the man moved 10 m, and the sand moves 6 m respect to him, then the sand moved 4 m for a man in the ground. So it would be -4

azizlwl

At beginnig both at origin.
1 second later, man moves to the right 10m
Now the man check that the sand is 6m behind him.
If the sand speed 16m/s, it will overtake the man.

If -4 means, 14m behind him.

Hernaner28

No, the sand moved 4 meters to the left so the velocity is -4.0 m/s. I don't know what's wrong with that

Hernaner28

[tex] \displaystyle {{v}_{2i}}\left( m+{{m}_{x}} \right)=\left( m-{{m}_{x}} \right){{v}_{2f}}+{{m}_{x}}\left( {{v}_{mass}} \right)[/tex]
[tex] \displaystyle 8\cdot 70=10\cdot 70-10{{m}_{x}}-{{m}_{x}}{{v}_{mass}}[/tex]

[tex] \displaystyle 8\cdot 70=10\cdot 70-10{{m}_{x}}-\left( -4{{m}_{x}} \right)[/tex]

[tex] \displaystyle {{m}_{x}}=23.33333[/tex]

Nothing was wrong with -4m/s!!!!

[tex]\displaystyle {{v}_{mass}}=-6.0+2.0=-4.0[/tex]

. I was mistaken with masses... not with relative speeds ¬¬