Mass Flux Fundamentals

Calculus of Variable Mass Systems

DOC ID: PHYS-VMS-001

DATE: ___________

Student Name

Course Section

Core Concept

Most introductory physics problems assume mass \(m\) is constant. In variable mass systems, we must consider the momentum flux—the rate at which mass enters or leaves the system boundaries. We use the fundamental definition: \(\mathbf{F}_{ext} = \frac{d\mathbf{p}}{dt}\).

1. Generalizing Newton's Second Law

Consider a system of mass \(M(t)\) moving with velocity \(v(t)\). A small mass \(dm\) is about to be added to the system, moving with velocity \(u\). After time \(dt\), the system has mass \(M + dm\) and moves with velocity \(v + dv\).

A) Write the total momentum of the system at time \(t\) and time \(t+dt\).

Momentum at \(t\): \(p(t)\)

Momentum at \(t+dt\): \(p(t+dt)\)

B) Show that the net external force can be expressed as: \(F_{ext} = M\frac{dv}{dt} - (u - v)\frac{dM}{dt}\)

2. Application: The Leaking Sand Bucket

A bucket of initial mass \(M_0\) is full of sand. It is pulled across a frictionless surface by a constant horizontal force \(P\). Sand leaks out of a hole in the bottom at a constant rate \(\mu = |dM/dt|\). The sand drops vertically out of the bucket.

Relative Velocity

What is \(u\) relative to the bucket for the leaking sand?

Mass at Time t

Express \(M(t)\) in terms of \(M_0\), \(\mu\), and \(t\).

Equation of Motion

Set up the differential equation for \(v(t)\).

C) Integrate to find the velocity \(v(t)\) as a function of time, assuming the bucket starts from rest.

3. Thinking Deeply: Momentum Flux and Paradoxes

Students often incorrectly apply \(F = \frac{d(mv)}{dt} = m\frac{dv}{dt} + v\frac{dm}{dt}\). Let's explore why this formula is specific to systems where the added mass comes from a rest frame (\(u=0\)) and is not universally true.

Case A: The Rain on the Cart

Vertical rain falls into a cart moving horizontally at velocity \(v\). The rain is initially at rest horizontally (\(u=0\)).

Internal System Force

F = m(dv/dt) + v(dm/dt)

Physical Reasoning

Explain why the cart must slow down if no external horizontal force is applied.

Case B: The Leaking Water Tanker

A tanker car leaks water vertically downward from its frame. The water leaves the tanker with the tanker's horizontal velocity \(v\).

Internal System Force

F = m(dv/dt)

Physical Reasoning

Explain why the velocity of the tanker remains constant if no external horizontal force is applied.

Critical Discussion Question

In Case B, mass is leaving the system. Why does the momentum of the tanker-plus-leaking-water system remain conserved while the velocity of the tanker itself doesn't change? Compare this to the rocket case where mass leaving the system causes a change in velocity.

"We must be careful of the definition of the system. Momentum is only conserved for a closed system." — Richard Feynman

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