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Variable mass systems
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Rockets and conveyor belts require the full dp/dt form.
F = m dv/dt + v dm/dtWhen mass enters or leaves a system, apply F = dp/dt carefully to a defined system including the added mass. Examples: a rocket, a leaking cart, rain falling in a moving wagon.
A coal-loaded belt needs extra force F = v dm/dt to accelerate new coal to belt speed.
Do not simply use m dv/dt without accounting for the mass change.
In more detail
In variable-mass systems, mass enters or leaves the object. The equation of motion must include the momentum carried by the mass that is gained or lost.
Real-world uses
- Rockets and jets lose mass as fuel is burned and expelled.
- Conveyor systems that pick up sand or gravel gain mass as they move.
- A conveyor belt that loads sand onto a moving surface must account for the mass added each second, so it cannot use F = ma directly.
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Frequently asked questions
Why can't I just use F = ma?
Because the mass changes with time, and the momentum carried by the escaping or incoming material matters.
Is this only for rockets?
No. It applies to any system where mass is added or removed, such as a chain falling off a table.
When does mass change in a system?
When material is added or lost, such as fuel burned by a rocket or sand dropped on a belt.
Related topics in Newton's Laws & Forces
Impulse · Linear momentum · Second law as rate of momentum change · Rocket propulsion · Conservation of momentum · Circular motion dynamics