doPhysics · Centre of Mass and Collisions
Momentum and Its Conservation
The momentum of a body and of a system, and the law that keeps the total fixed when no net external force acts. Explosions and recoil, and momentum along one direction with energy alongside. Bodies whose mass changes as they move: rockets, sand on a belt and a falling chain.
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Why this chapter
Why this chapter exists
Newton's laws follow each body force by force. In a burst, a recoil or a crash, the forces are huge, brief and unknown, and that method has nothing to work with. Momentum offers a way round. The forces inside a system come in equal and opposite pairs, so they can never change its total momentum. That one fact gives the speeds after an explosion without knowing a single force. This chapter builds the law and uses it one direction at a time alongside energy. Then it extends the law to rockets and streams whose mass changes.
You will learn
By the end of this chapter you will
- State the second law as \(\vec F = d\vec p/dt\), and explain why \(\vec F = m\vec a\) is its constant-mass case.
- Find the total momentum of a system as a vector sum, and relate it to the velocity of the centre of mass.
- Explain why internal forces cannot change the total momentum, and decide when momentum is conserved: fully, along one direction, or across a brief event.
- Solve explosions, the recoil of a gun and a person leaving a cart, with velocities measured over the ground.
- Combine horizontal momentum with energy for a block on a free wedge or in a free groove, and for a bead with a bob.
- Derive the thrust of a rocket and the rocket equation \(v = u\ln(m_0/m)\), with and without gravity.
- Find the force of a steady stream, the power needed for a sand belt, and the push of a table under a falling chain.
Before you begin
Before you begin
From the chapter on Accelerated Objects, you should know momentum \(\vec p = m\vec v\) and the impulse-momentum theorem \(\vec J = \Delta\vec p\). You should know Newton's third law, the conservation of mechanical energy, and relative velocity along a line. You should be able to differentiate a product and integrate \(1/x\).
Lessons
1
Momentum of a System
Momentum recalled, and the second law in its general form, force as the rate of change of momentum. The total momentum of a system of bodies, and the one rule that changes it, the net external force.
Momentum and the general second lawTotal momentum of a systemExternal force changes total momentum
2
The Law of Conservation of Momentum
With no net external force the total momentum of a system cannot change. The law in full, and the law for one component alone. The law across a brief event, when the internal forces far exceed the external ones.
Internal forces cannot change total momentumThe law of conservation of momentumConservation along one directionConservation during a brief interval
3
Explosions and Recoil
A body that bursts into two or three pieces. The recoil of a gun, and why the bullet takes almost all the energy. A gun on a wagon that fires twice, and a person who leaves a cart at a speed measured relative to it.
A body that bursts into piecesRecoil of a gunJumping off a cart
4
Momentum Along One Direction
Bodies that move on or inside a free block on a smooth floor. Horizontal momentum is conserved while vertical momentum is not, and energy supplies the second equation. A block on a free wedge, a block in a free groove, and a bead on a wire with a bob hanging from it.
Momentum and energy togetherBlock in the groove of a free blockBead on a wire and a hanging bob
5
Rockets and Thrust
Systems that lose or gain mass. The momentum account over a short time and the thrust of the gas that leaves. The rocket equation \(v = u \ln(m_0/m)\), and the burn against gravity.
Momentum account of an open systemThrust and the equation of motionThe rocket equationA rocket rising against gravity
6
Sand on a Belt and a Falling Chain
Continuous streams of mass that strike, land or are carried along. The force of a jet on a wall, and the power needed for a belt under falling sand. Why a chain falling onto a table presses with three times the landed weight.
Force of a steady streamSand on a moving beltA chain falling on a table
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
0%The Workbook
Questions to solve on paper, marked by you.
Glossary
The key words of this chapter, with the meaning this chapter gives them.
- centre of mass
- The mass-weighted average position of the parts of a system. The total momentum of the system is its total mass times the velocity of this point.
- closed system
- A system that keeps the same particles all the time. The second law \(\vec F_{\text{ext}} = d\vec P/dt\) applies to a closed system.
- exhaust speed
- The speed of the gas leaving a rocket, measured relative to the rocket. The engine fixes it, whatever the speed of the rocket.
- explosion
- A sudden internal event, lasting milliseconds, that drives the parts of a body apart. Momentum is conserved across it, while kinetic energy grows.
- external force
- A force on a part of a system from a body outside it, such as gravity or the push of a floor. Only external forces change the total momentum.
- heat
- Energy in the random motion of the molecules of a body. Friction turns kinetic energy into heat, as when a grain of sand slips on a moving belt.
- impulse
- The product of a force and the time for which it acts, or the area under the force-time graph. The impulse of the net force equals the change in momentum, in N s.
- internal force
- A force that one part of a system exerts on another part of the same system. Internal forces come in equal and opposite third-law pairs.
- isolated system
- A system on which the net external force is zero. Its total momentum stays constant.
- mass flow rate
- The mass of a stream arriving at, or leaving, a surface each second, \(\mu = dm/dt\), in kg/s.
- mass ratio
- The mass of a rocket at the start of a burn divided by its mass at the end, \(m_0/m\). The speed gained is the exhaust speed times its natural logarithm.
- momentum
- The mass of a body times its velocity, \(\vec p = m\vec v\). It is a vector along the velocity, measured in kg m/s.
- muzzle velocity
- The velocity of a bullet or a shell relative to the gun that fires it. Its velocity over the ground is the muzzle velocity plus the gun's own velocity.
- open system
- A system that gains or loses mass, such as a rocket burning fuel or a belt collecting sand. The mass that crosses its boundary carries momentum with it.
- recoil
- The backward motion of a gun, a cannon or a person when a bullet, a ball or a push is sent forward. The recoil momentum equals the forward momentum.
- system
- The set of bodies chosen for study, marked off by a boundary. Every force on a part of it is either internal or external.
- thrust
- The push on a body from the mass it throws out or takes in, \(v_{\text{rel}}\,dm/dt\). For a rocket it is the exhaust speed times the rate of burning.