Rocket equation: leaking bucket

In summary, the problem involves a bucket of mass M being pulled up a well by a rope while containing a mass of water. The water is leaking out at a steady rate, causing the bucket to become empty after a certain time T. The question asks for the velocity of the bucket at the moment it becomes empty. The key concept is that the relative velocity of the water to the bucket is 0, but this only lasts for an instant as the bucket is accelerating. This is similar to an aircraft dropping a bomb, where the bomb initially has the same velocity as the aircraft.
  • #1
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Homework Statement


A bucket of mass M (when empty) initial at rest and containing a mass of water is being pulled up a well by a rope exerting a steady force P. The water is leaking out of the bucket at a steady rate such that the bucket is empty after a time T. Find the velocity of the bucket at the instant it becomes empty.


Homework Equations


rocket equation:
Fext = m dv/dt - Vrel dm/dt

The Attempt at a Solution


the total mass of the bucket and water is M' = M + m - mt/T , where m is the initial mass of the water.
The hint said " as the leaking water has zero velocity relative to the bucket.."

I don't understand it. Why Water should have zero velocity relative to the bucket?? Water is leaking out at a steady rate, so there should be a non-zero constant velocity. i am so confused.
 
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  • #2
It is just like a aircraft drop a bomb when it flies. The bomb will move with the same speed as the aircraft.

The relative velocity of water relative to the bucket is 0, however in this question,the bucket is accelerating so the zero relative velocity only last for an instant.
 

1. What is the rocket equation?

The rocket equation is a mathematical formula that describes the motion of a rocket based on the principles of Newton's laws of motion. It takes into account the mass of the rocket, the mass of the propellant, and the velocity of the exhaust gases to calculate the velocity and acceleration of the rocket.

2. How does the rocket equation work?

The rocket equation uses the conservation of momentum and the conservation of mass to calculate the change in velocity of a rocket. It states that for every action (exhaust gases being expelled from the rocket) there is an equal and opposite reaction (the rocket moving in the opposite direction).

3. What is the significance of the "leaking bucket" in the rocket equation?

The "leaking bucket" analogy is used to explain the concept of the rocket equation. Just as a leaking bucket will become lighter and lighter as water leaks out, a rocket will become lighter as it expels propellant. This decrease in mass leads to an increase in velocity according to the rocket equation.

4. How does the rocket equation impact space exploration?

The rocket equation is essential in understanding and designing rockets for space exploration. It helps scientists and engineers determine the amount of propellant needed for a specific mission and the maximum velocity a rocket can achieve. It also highlights the importance of reducing the mass of a rocket to increase its efficiency.

5. Are there any limitations to the rocket equation?

The rocket equation assumes that the rocket's velocity is constant throughout the burn time and that all the propellant is exhausted at once. In reality, the velocity of a rocket changes continuously, and the propellant is used up gradually. These limitations can be accounted for by using more complex equations, such as the Tsiolkovsky rocket equation, which takes into account the changing mass of the rocket during the burn time.

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