Mechanics with mass as a function of time

[hyrodi]

Homework Statement

Suppose there is a radioactive container at rest on a frictionless plane in a vacuum. Due to faulty manufacturers who skimped on welding and opted for duct tape, it is leaking material at three kilograms per minute. Several mischievous gnomes sneak out and begin to push the container to the right with a constant force of 15N. At the moment the gnomes begin to push the total mass of the container is 133 kg. What is the velocity of the container three seconds after the gnomes begin pushing?

Homework Equations

F = ma
m = f(t)

The Attempt at a Solution

Converting from 3 kg/min to seconds; 3/60 = 1/20;
mass = (133 - t/20) kg

so F = ma, 15N = (133 - t/20)(a)

a = 15 / (133 - t/20)

Integrate from 0 to 3 seconds, velocity = 0.339

Did I get it right? It is correct, when treating mass as a function of time, to solve questions of this type this way?

Thank you for any assistance. :)

 

[anathema]

Hello there,

Thank you for your question. Your solution is correct, but there are a few things to consider.

First, when solving problems involving forces and motion, it is important to consider all the forces acting on the object. In this case, there are two forces acting on the container: the force of the gnomes pushing it to the right, and the force of the leaking material pushing it to the left. So the equation should be F = ma = Fgnomes - Fleak, where Fgnomes is the force of the gnomes and Fleak is the force of the leaking material.

Second, you can simplify the problem by treating the mass as a constant rather than a function of time. Since the leaking material is constant at 3 kg/min, the mass of the container will decrease by 3 kg every minute. So after 3 seconds, the mass of the container will be 133 kg - (3 kg/min)(3 s)(1 min/60 s) = 132.45 kg. This is close enough to the original mass of 133 kg that we can treat it as a constant for this problem.

So the equation becomes F = ma = 15 N - (3 kg/min)(1 min/60 s) = 14.95 N. Solving for acceleration, a = F/m = 14.95 N / 132.45 kg = 0.113 m/s^2.

Finally, to find the velocity after 3 seconds, we can use the equation v = u + at, where u is the initial velocity (which is 0 since the container is at rest) and t is the time. So v = (0 m/s) + (0.113 m/s^2)(3 s) = 0.339 m/s.

So your final answer is correct, but there are some small details to consider. I hope this helps, and let me know if you have any other questions. Good luck with your studies!