Linear Momentum

4.1

Syllabus

Learning Objective	Essential Knowledge
4.1.A Describe the linear momentum of an object or system.	4.1.A.1 Linear momentum is defined by the equation $\vec{p} = m\vec{v}$. 4.1.A.2 Momentum is a vector quantity and has the same direction as the velocity. 4.1.A.3 Momentum can be used to analyze collisions and explosions. 4.1.A.3.i A collision is a model for an interaction where the forces exerted between the involved objects in the system are much larger than the net external force exerted on those objects during the interaction. 4.1.A.3.ii As only the initial and final states of a collision are analyzed, the object model may be used to analyze collisions. 4.1.A.3.iii An explosion is a model for an interaction in which forces internal to the system move objects within that system apart.

Learning Objective

Essential Knowledge

4.1.A
Describe the linear momentum of an object or system.

4.1.A.1 Linear momentum is defined by the equation $\vec{p} = m\vec{v}$.
4.1.A.2 Momentum is a vector quantity and has the same direction as the velocity.
4.1.A.3 Momentum can be used to analyze collisions and explosions.
- 4.1.A.3.i A collision is a model for an interaction where the forces exerted between the involved objects in the system are much larger than the net external force exerted on those objects during the interaction.
- 4.1.A.3.ii As only the initial and final states of a collision are analyzed, the object model may be used to analyze collisions.
- 4.1.A.3.iii An explosion is a model for an interaction in which forces internal to the system move objects within that system apart.

Source: College Board AP Course and Exam Description

Linear momentum 动量 is mass times velocity:

$$\vec{p}=m\vec{v}.$$

It is a vector 矢量 pointing along the velocity, and it is the quantity for analysing collisions 碰撞 and explosions 爆炸. A collection of objects can be treated as one system moving with the velocity of its center of mass 质心:

$$\vec{v}_{\text{cm}}=\frac{\sum m_i\vec{v}_i}{\sum m_i},$$

so the system's total momentum is its total mass times $\vec{v}_{\text{cm}}$ – one object's worth of bookkeeping for any number of parts.

Vocabulary Train

English	Chinese	Pinyin
Linear momentum	动量	dòng liàng
vector	矢量	shǐ liàng
collisions	碰撞	pèng zhuàng
explosions	爆炸	bào zhà
center of mass	质心	zhì xīn

Exercise sheet

4.2

Change in Momentum and Impulse

Syllabus

Learning Objective	Essential Knowledge
4.2.A Describe the impulse delivered to an object or system.	4.2.A.1 The rate of change of a system's momentum is equal to the net external force exerted on that system. Equation: $\vec{F}_{\text{net}} = \dfrac{d\vec{p}}{dt}$ 4.2.A.2 Impulse is defined as the integral of a force exerted on an object or system over a time interval. Equation: $\vec{J} = \displaystyle\int_{t_1}^{t_2} \vec{F}_{\text{net}}(t)\,dt$ 4.2.A.3 Impulse is a vector quantity and has the same direction as the net force exerted on the system. 4.2.A.4 The impulse delivered to a system by a net external force is equal to the area under the curve of a graph of the net external force exerted on the system as a function of time. 4.2.A.5 The net external force exerted on a system is equal to the slope of a graph of the momentum of the system as a function of time.
4.2.B Describe the relationship between the impulse exerted on an object or system and the change in momentum of the object or system.	4.2.B.1 Change in momentum is the difference between a system's final momentum and its initial momentum. Equation: $\Delta\vec{p} = \vec{p} - \vec{p}_0$ 4.2.B.2 The impulse–momentum theorem relates the impulse delivered to an object and the object's change in momentum. 4.2.B.2.i The impulse exerted on an object is equal to the object's change in momentum. Equation: $\vec{J} = \displaystyle\int_{t_1}^{t_2} \vec{F}_{\text{net}}(t)\,dt = \Delta\vec{p}$ 4.2.B.2.ii Newton's second law of motion is a direct result of the impulse–momentum theorem applied to systems with constant mass. Equation: $\vec{F}_{\text{net}} = \dfrac{d\vec{p}}{dt} = m\dfrac{d\vec{v}}{dt} = m\vec{a}$ 4.2.B.2.iii The impulse–momentum theorem also describes the behavior of a system in which the velocity is constant but the mass changes with respect to time. Equation: $\vec{F}_{\text{net}} = \dfrac{d\vec{p}}{dt} = \dfrac{dm}{dt}\vec{v}$

Learning Objective

Essential Knowledge

4.2.A
Describe the impulse delivered to an object or system.

4.2.A.1 The rate of change of a system's momentum is equal to the net external force exerted on that system.
- Equation: $\vec{F}_{\text{net}} = \dfrac{d\vec{p}}{dt}$
4.2.A.2 Impulse is defined as the integral of a force exerted on an object or system over a time interval.
- Equation: $\vec{J} = \displaystyle\int_{t_1}^{t_2} \vec{F}_{\text{net}}(t)\,dt$
4.2.A.3 Impulse is a vector quantity and has the same direction as the net force exerted on the system.
4.2.A.4 The impulse delivered to a system by a net external force is equal to the area under the curve of a graph of the net external force exerted on the system as a function of time.
4.2.A.5 The net external force exerted on a system is equal to the slope of a graph of the momentum of the system as a function of time.

4.2.B
Describe the relationship between the impulse exerted on an object or system and the change in momentum of the object or system.

4.2.B.1 Change in momentum is the difference between a system's final momentum and its initial momentum.
- Equation: $\Delta\vec{p} = \vec{p} - \vec{p}_0$
4.2.B.2 The impulse–momentum theorem relates the impulse delivered to an object and the object's change in momentum.
- 4.2.B.2.i The impulse exerted on an object is equal to the object's change in momentum.
  - Equation: $\vec{J} = \displaystyle\int_{t_1}^{t_2} \vec{F}_{\text{net}}(t)\,dt = \Delta\vec{p}$
- 4.2.B.2.ii Newton's second law of motion is a direct result of the impulse–momentum theorem applied to systems with constant mass.
  - Equation: $\vec{F}_{\text{net}} = \dfrac{d\vec{p}}{dt} = m\dfrac{d\vec{v}}{dt} = m\vec{a}$
- 4.2.B.2.iii The impulse–momentum theorem also describes the behavior of a system in which the velocity is constant but the mass changes with respect to time.
  - Equation: $\vec{F}_{\text{net}} = \dfrac{d\vec{p}}{dt} = \dfrac{dm}{dt}\vec{v}$

Source: College Board AP Course and Exam Description

Newton's second law is really a statement about momentum:

$$\vec{F}_{\text{net}}=\frac{d\vec{p}}{dt},$$

which reduces to $m\vec{a}$ when the mass is constant – and handles a changing mass ($\vec{F}=\dfrac{dm}{dt}\vec{v}$, a rocket or a chain piling onto a scale) when it is not. The impulse 冲量 delivered by a force is its time integral, and the impulse–momentum theorem 冲量-动量定理 says it equals the change in momentum:

$$\vec{J}=\int_{t_1}^{t_2}\vec{F}_{\text{net}}\,dt=\Delta\vec{p}.$$

Impulse is a vector along the net force. Read it off graphs both ways:

On a force–time graph, impulse is the area under the curve.
On a momentum–time graph, the net force is the slope 斜率 at each instant.

Impulse is the area under the force-time curve, equal to the average force times the contact time

Spreading the same $\Delta p$ over a longer time lowers the force – that is why airbags, crumple zones, and soft landings work, and why you bend your knees when you land.

Worked example. A time-varying force $F(t)=10t\ \text{N}$ acts on a $2.0\ \text{kg}$ object for $2.0\ \text{s}$. The impulse is $J=\displaystyle\int_0^2 10t\,dt=\big[5t^2\big]_0^2=20\ \text{N}\cdot\text{s}$, so the speed changes by $\Delta v=J/m=10\ \text{m/s}$.

Vocabulary Train

English	Chinese	Pinyin
impulse	冲量	chōng liàng
impulse–momentum theorem	冲量-动量定理	chōng liàng - dòng liàng dìng lǐ
slope	斜率	xié lǜ

Exercise sheet

4.3

Conservation of Linear Momentum

Syllabus

Learning Objective	Essential Knowledge
4.3.A Describe the behavior of a system using conservation of linear momentum.	4.3.A.1 A collection of objects with individual momenta can be described as one system with one center-of-mass velocity. 4.3.A.1.i For a collection of objects, the velocity of a system's center of mass can be calculated using the equation Equation: $\vec{v}_{\text{cm}} = \dfrac{\sum \vec{p}_i}{\sum m_i} = \dfrac{\sum (m_i \vec{v}_i)}{\sum m_i}$ 4.3.A.1.ii The velocity of a system's center of mass is constant in the absence of a net external force. 4.3.A.2 The total momentum of a system is the sum of the momenta of the system's constituent parts. 4.3.A.3 In the absence of net external forces, any change to the momentum of an object within a system must be balanced by an equivalent and opposite change of momentum elsewhere within the system. Any change to the momentum of a system is due to a transfer of momentum between the system and its surroundings. 4.3.A.3.i The impulse exerted by one object on a second object is equal and opposite to the impulse exerted by the second object on the first. This is a direct result of Newton's third law. 4.3.A.3.ii A system may be selected so that the total momentum of that system is constant. 4.3.A.3.iii If the total momentum of a system changes, that change will be equivalent to the impulse exerted on the system. Equation: $\vec{J} = \Delta\vec{p}$ 4.3.A.4 Correct application of conservation of momentum can be used to determine the velocity of a system immediately before and immediately after collisions or explosions. Boundary statement: AP Physics C: Mechanics only expects students to quantitatively analyze collisions and interactions in one or two dimensions. Three-dimensional collisions may be analyzed qualitatively.
4.3.B Describe how the selection of a system determines whether the momentum of that system changes.	4.3.B.1 Momentum is conserved in all interactions. 4.3.B.2 If the net external force on the selected system is zero, the total momentum of the system is constant. 4.3.B.3 If the net external force on the selected system is nonzero, momentum is transferred between the system and the environment.

Learning Objective

Essential Knowledge

4.3.A
Describe the behavior of a system using conservation of linear momentum.

4.3.A.1 A collection of objects with individual momenta can be described as one system with one center-of-mass velocity.
- 4.3.A.1.i For a collection of objects, the velocity of a system's center of mass can be calculated using the equation
  - Equation: $\vec{v}_{\text{cm}} = \dfrac{\sum \vec{p}_i}{\sum m_i} = \dfrac{\sum (m_i \vec{v}_i)}{\sum m_i}$
- 4.3.A.1.ii The velocity of a system's center of mass is constant in the absence of a net external force.
4.3.A.2 The total momentum of a system is the sum of the momenta of the system's constituent parts.
4.3.A.3 In the absence of net external forces, any change to the momentum of an object within a system must be balanced by an equivalent and opposite change of momentum elsewhere within the system. Any change to the momentum of a system is due to a transfer of momentum between the system and its surroundings.
- 4.3.A.3.i The impulse exerted by one object on a second object is equal and opposite to the impulse exerted by the second object on the first. This is a direct result of Newton's third law.
- 4.3.A.3.ii A system may be selected so that the total momentum of that system is constant.
- 4.3.A.3.iii If the total momentum of a system changes, that change will be equivalent to the impulse exerted on the system.
  - Equation: $\vec{J} = \Delta\vec{p}$
4.3.A.4 Correct application of conservation of momentum can be used to determine the velocity of a system immediately before and immediately after collisions or explosions.

Boundary statement: AP Physics C: Mechanics only expects students to quantitatively analyze collisions and interactions in one or two dimensions. Three-dimensional collisions may be analyzed qualitatively.

4.3.B
Describe how the selection of a system determines whether the momentum of that system changes.

4.3.B.1 Momentum is conserved in all interactions.
4.3.B.2 If the net external force on the selected system is zero, the total momentum of the system is constant.
4.3.B.3 If the net external force on the selected system is nonzero, momentum is transferred between the system and the environment.

Source: College Board AP Course and Exam Description

Internal forces come in Newton's-third-law pairs, so they cancel inside any system: they can shuffle momentum between parts but never change the total. Momentum only enters or leaves a system through a net external force ($\vec{J}=\Delta\vec{p}$). So, with zero net external force, total momentum is conserved 守恒:

$$\sum\vec{p}_{\text{before}}=\sum\vec{p}_{\text{after}}.$$

Momentum is conserved in every collision, however violent – choose the system large enough that the collision forces are internal. Apply conservation separately along each axis; AP asks for quantitative work in one or two dimensions.

A head-on collision: total momentum before equals total momentum after

Worked example (explosion). A $6.0\ \text{kg}$ shell at rest splits into a $2.0\ \text{kg}$ piece moving at $9.0\ \text{m/s}$ east and a $4.0\ \text{kg}$ piece. Total momentum stays zero, so the heavy piece moves west at $v=\dfrac{2.0(9.0)}{4.0}=4.5\ \text{m/s}$. The kinetic energy came from stored (chemical or spring) energy – momentum conservation does not require kinetic-energy conservation.

Explore

Collide two carts and conserve momentum

In any collision the total momentum $\sum mv$ before equals the total after. Set the masses and speeds and check the momentum bookkeeping.

Vocabulary Train

English	Chinese	Pinyin
conserved	守恒	shǒu héng

Exercise sheet

4.4

Elastic and Inelastic Collisions

Syllabus

Learning Objective	Essential Knowledge
4.4.A Describe whether an interaction between objects is elastic or inelastic.	4.4.A.1 An elastic collision between objects is one in which the initial kinetic energy of the system is equal to the final kinetic energy of the system. 4.4.A.2 In an elastic collision, the final kinetic energies of each of the objects within the system may be different from their initial kinetic energies. 4.4.A.3 An inelastic collision between objects is one in which the total kinetic energy of the system decreases. 4.4.A.4 In an inelastic collision, some of the initial kinetic energy is not restored to kinetic energy but is transformed by nonconservative forces into other forms of energy. 4.4.A.5 In a perfectly inelastic collision, the objects stick together and move with the same velocity after the collision.

Learning Objective

Essential Knowledge

4.4.A
Describe whether an interaction between objects is elastic or inelastic.

4.4.A.1 An elastic collision between objects is one in which the initial kinetic energy of the system is equal to the final kinetic energy of the system.
4.4.A.2 In an elastic collision, the final kinetic energies of each of the objects within the system may be different from their initial kinetic energies.
4.4.A.3 An inelastic collision between objects is one in which the total kinetic energy of the system decreases.
4.4.A.4 In an inelastic collision, some of the initial kinetic energy is not restored to kinetic energy but is transformed by nonconservative forces into other forms of energy.
4.4.A.5 In a perfectly inelastic collision, the objects stick together and move with the same velocity after the collision.

Source: College Board AP Course and Exam Description

Conservation of momentum in a collision

All collisions conserve momentum; they differ in what happens to the kinetic energy 动能:

type	momentum	kinetic energy
elastic collision 弹性碰撞	conserved	total conserved (individual shares may change)
inelastic collision 非弹性碰撞	conserved	decreases – some becomes heat, sound, deformation 形变
perfectly inelastic collision 完全非弹性碰撞	conserved	largest possible loss – the objects stick and share one velocity

Strategy: always write momentum conservation first; add the kinetic-energy equation only when the problem says "elastic". Two useful elastic facts: equal masses in a 1D elastic collision simply exchange velocities, and in the center-of-mass frame each object just reverses its velocity.

Worked example. A $1000\ \text{kg}$ car at $20\ \text{m/s}$ strikes a stationary $1500\ \text{kg}$ car and they lock together – perfectly inelastic. Momentum: $v=\dfrac{1000(20)}{2500}=8.0\ \text{m/s}$. Kinetic energy falls from $2.0\times10^5\ \text{J}$ to $\tfrac12(2500)(8.0)^2=8.0\times10^4\ \text{J}$: about $60\%$ is lost, even though momentum is exactly conserved.

Worked example (2D). A puck moving east at $4.0\ \text{m/s}$ strikes an identical puck at rest; after the glancing hit, one moves at $2.0\ \text{m/s}$ at $60^\circ$ north of east. Conserve each axis: east–west, $m(4.0)=m(2.0)\cos60^\circ+mv_x$, so $v_x=3.0\ \text{m/s}$; north–south, $0=m(2.0)\sin60^\circ-mv_y$, so $v_y=1.7\ \text{m/s}$. The second puck moves at $\sqrt{3.0^2+1.7^2}=3.5\ \text{m/s}$, about $30^\circ$ south of east.

A glancing collision, resolved along two perpendicular axes

Exam skill. On FRQs, justify with the condition, not the slogan: "the net external force on the two-puck system is zero during the collision, so its total momentum is constant." If asked whether the collision is elastic, compute the kinetic energy before and after and compare – never assume.

A Newton's cradle: five steel balls hanging in a row A Newton's cradle shows momentum and kinetic energy passing through a near-elastic collision

Explore

Compare elastic and inelastic collisions

Momentum is always conserved, but kinetic energy is only conserved in an elastic collision. In an inelastic one the carts stick and some energy becomes heat.

Vocabulary Train

English	Chinese	Pinyin
kinetic energy	动能	dòng néng
elastic collision	弹性碰撞	tán xìng pèng zhuàng
inelastic collision	非弹性碰撞	fēi tán xìng pèng zhuàng
deformation	形变	xíng biàn
perfectly inelastic collision	完全非弹性碰撞	wán quán fēi tán xìng pèng zhuàng

Exercise sheet

4.4

Exam tips

Impulse equals the momentum change: $\vec J=\int \vec F\,dt=\Delta\vec p$, and it is the area under a force–time graph.
Momentum is conserved whenever the net external force is zero — always the go-to for collisions.
Distinguish elastic (kinetic energy conserved) from inelastic (objects stick) collisions.
Apply conservation to each component (x and y) separately in 2-D.
Connect to center of mass: total momentum $=M\vec v_{cm}$.

Linear Momentum

Linear Momentum

Change in Momentum and Impulse

Conservation of Linear Momentum

Collide two carts and conserve momentum

Elastic and Inelastic Collisions

Compare elastic and inelastic collisions

Exam tips

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