A Physics Question

[匿名]

2009-08-27 4:57 am

Container A and B contains some water.
They are connected with a rope that pass through a pulley.
The mass of both containers are m kg(including the water inside), and
the mass of water inside the container is 0.5m kg.
The whole system is initially at rest.
But at t=0, a hole is made under container A, water inside container A
start to flow out of the container through the hole, and container A
begin to move upward. It is given that the instantaneous mass of water
that is flowing out through the hole at time t equals to kt (where k is a constant).
Find the velocity of container A when all the water in container A
have flowed out.

(Please express your answer in terms of m, g and k. And show your
steps clearly)

Thanks.

更新1:

I am sorry that I have not explained it clearly. I mean the mass of both container and the water are 0.5m kg.

更新2:

Also, I want to ask a question, how to integrate the following functions? (a) f(x)=(k-x)/(k+x) (k is a constant) (b) f(x)=(kgx^2)/(4m-kx^2) (k, g and m are constant) Please show the steps clearly with simple elaborations. Thanks.

回答 (1)

Prof. Physics

2009-08-27 5:16 am

✔ 最佳答案

At time = t, mass of container A = (m + 0.5m) - int (0 to t) kt dt

= 1.5m - kt2/2

By Newton's 2nd law of motion,

For B, (m + 0.5m)dv/dt = (m + 0.5m)g - T

1.5mg - T = 1.5mdv/dt ... (1)

For A, (1.5m - kt2/2)dv/dt = T - (1.5m - kt2/2)g ... (2)

(1) + (2):

(3m - kt2/2)dv/dt = gkt2/2

int dv = int (gkt2/2)/(3m - kt2/2)dt

-v/g = int [1 - 6m/(6m - kt2)] dt

-v/g = t - sqrt(6m/k) ln[(sqrt(6m/k) + t) / sqrt(6m/k - t2)] + C

When t = 0, v = 0, C = 0

When the time that all water in A has flowed out,

t = sqrt(m/k)

-v/g = sqrt(m/k) - 1/2 sqrt(6m/k) ln(7/5)

Speed of A, v = g sqrt(m/k) [sqrt(6)/2 X ln(7/5) - 1]

2009-08-28 18:34:14 補充：
Then, you will need some fine adjustment in my calculation.
But the outline of my calculation is correct. You just amend the values.

2009-08-28 18:35:02 補充：
About how to integrate those functions, these are not physics questions.
But I will give you some ideas to do it.
the first one, maybe you need partial fraction.
the second one, you will need substitution of trigo. functions to do it.

參考: Physics king

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