Conservation of Momentum with Varying Mass

In summary, we are trying to find the velocity and position of a boat with mass M, which is at rest. Balls with mass of m are being thrown continuously at the back of the boat at a rate of σ kg/s, and are being collected inside the boat through inelastic collisions. Using the equation Δmu + MV = (Δm + M)(V + ΔV), we can assume that linear momentum is conserved as there are no outer forces acting on the system. Therefore, we can use the rocket equation to calculate the velocity and position of the boat over time.
  • #1
godman1402
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Homework Statement



A boat with mass M is at rest. Balls are thrown at the back of the boat, where each ball has mass of m, and the balls are being thrown with the mass rate of σ kg/s (the rate is continuous). The balls are being collected inside the boat (inelastic collision). Find velocity and position of the boat as a function of time.

Homework Equations



Δmu + MV = (Δm + M)(V + ΔV)

The Attempt at a Solution



I thought I could use the rocket equation for this, but I had trouble figuring it out because I wasn't sure if I can assume ƩF = 0.
 
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  • #2
Rockets don't usually scoop up their own exhaust products. Once the exhaust mass leaves the rocket, it is assumed to be lost permanently.
 
  • #3
Yes you can go ahead and use the conservation, there is no outer force acting on the system. The Boat is not fixed by some outer force so we can translate it in space and get the same dynamics meaning linear momentum is conserved (we assume the effect of water on the boat negligible).
 

1. What is conservation of momentum with varying mass?

Conservation of momentum with varying mass is a fundamental principle in physics that states that the total momentum of a system remains constant if there are no external forces acting on it. This means that the combined momentum of all objects in the system, even if their masses are changing, will remain the same.

2. How is conservation of momentum with varying mass related to Newton's third law?

Conservation of momentum with varying mass is closely related to Newton's third law of motion, which states that for every action, there is an equal and opposite reaction. In a system with varying mass, the change in momentum of one object is equal to the change in momentum of another object in the opposite direction, allowing for the total momentum to remain constant.

3. Can conservation of momentum with varying mass be applied to all types of systems?

Yes, conservation of momentum with varying mass is a universal principle that can be applied to all types of systems, whether they are macroscopic objects or subatomic particles. This principle is particularly useful in analyzing collisions and other interactions between objects.

4. How does conservation of momentum with varying mass affect the motion of objects?

Conservation of momentum with varying mass plays a crucial role in determining the motion of objects in a system. It allows for the calculation of velocities and directions of objects after a collision or interaction, as well as predicting the overall motion of the system as a whole.

5. Are there any real-life applications of conservation of momentum with varying mass?

Yes, conservation of momentum with varying mass has numerous real-life applications. It is used in engineering and design of vehicles and structures, as well as in understanding the behavior of celestial bodies in orbit. It also plays a crucial role in sports, such as in the trajectory of a ball after being hit or thrown.

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