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 $\vec{p} = m\vec{v}$. 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. Boundary statement: Unless otherwise stated, the general term "momentum" will refer specifically to linear momentum.

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

Boundary statement: Unless otherwise stated, the general term "momentum" will refer specifically to linear momentum.

Source: College Board AP Course and Exam Description

Linear momentum 动量 is mass times velocity – a vector pointing the same way as the velocity:

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

It measures "how hard it is to stop" a moving object. A heavy slow truck and a light fast ball can have the same momentum. Its unit is $\text{kg m/s}$, the same as $\text{N s}$.

Vocabulary Train

English	Chinese	Pinyin
Linear momentum	动量	dòng liàng

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 momentum is equal to the net external force exerted on an object or system. Equation: $\vec{F}_{\text{net}} = \dfrac{\Delta \vec{p}}{\Delta t}$ 4.2.A.2 Impulse is defined as the product of the average force exerted on a system and the time interval during which that force is exerted on the system. Equation: $\vec{J} = \vec{F}_{\text{avg}} \Delta t$ 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 a 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 exerted on a system and the system's change in momentum. Equation: $\vec{J} = \vec{F}_{\text{avg}} \Delta t = \Delta \vec{p}$ 4.2.B.3 Newton's second law of motion is a direct result of the impulse–momentum theorem applied to systems with constant mass. Equation: $\vec{F}_{\text{net}} = \dfrac{\Delta \vec{p}}{\Delta t} = m\dfrac{\Delta \vec{v}}{\Delta t} = m\vec{a}$ Boundary statement: AP Physics 1 does not require students to quantitatively analyze systems in which the mass of the system changes with respect to time.

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 momentum is equal to the net external force exerted on an object or system.
- Equation: $\vec{F}_{\text{net}} = \dfrac{\Delta \vec{p}}{\Delta t}$
4.2.A.2 Impulse is defined as the product of the average force exerted on a system and the time interval during which that force is exerted on the system.
- Equation: $\vec{J} = \vec{F}_{\text{avg}} \Delta t$
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 a 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 exerted on a system and the system's change in momentum.
- Equation: $\vec{J} = \vec{F}_{\text{avg}} \Delta t = \Delta \vec{p}$
4.2.B.3 Newton's second law of motion is a direct result of the impulse–momentum theorem applied to systems with constant mass.
- Equation: $\vec{F}_{\text{net}} = \dfrac{\Delta \vec{p}}{\Delta t} = m\dfrac{\Delta \vec{v}}{\Delta t} = m\vec{a}$

Boundary statement: AP Physics 1 does not require students to quantitatively analyze systems in which the mass of the system changes with respect to time.

Source: College Board AP Course and Exam Description

A net force acting over time changes momentum. The impulse 冲量 delivered is

$$\vec{J}=\vec{F}\,\Delta t=\Delta\vec{p}.$$

This is the impulse–momentum theorem: impulse equals the change in momentum. On a force–time graph, impulse is the area under the curve. It explains why airbags and follow-through help – spreading the same momentum change over a longer time reduces the force.

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

Worked example. A $0.15\ \text{kg}$ ball hits a wall at $20\ \text{m/s}$ and bounces straight back at $15\ \text{m/s}$. The contact lasts $0.020\ \text{s}$. Find the average force on the ball. Take the rebound direction as positive, so $u=-20\ \text{m/s}$ and $v=+15\ \text{m/s}$:

$$\Delta p=m(v-u)=0.15\big(15-(-20)\big)=5.25\ \text{kg m/s},\qquad F=\frac{\Delta p}{\Delta t}=\frac{5.25}{0.020}=260\ \text{N}.$$

The sign work matters: forgetting that the velocity reverses is the most common mistake here.

Vocabulary Train

English	Chinese	Pinyin
impulse	冲量	chōng liàng

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 $\vec{v}_{\text{cm}} = \dfrac{\sum \vec{p}_i}{\sum m_i} = \dfrac{\sum m_i \vec{v}_i}{\sum m_i}$. 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 1 includes a quantitative and qualitative treatment of conservation of momentum in one dimension and a semiquantitative treatment of conservation of momentum in two dimensions. Exam questions involving solution of simultaneous equations are not included in AP Physics 1, but the AP Physics 1 Exam may include questions that assess whether students can set up the equations properly and reason about how changing a given mass, speed, or angle would affect other quantities. AP Physics 2 includes a full treatment of conservation of momentum in two dimensions for problems that include one unknown final velocity.
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 $\vec{v}_{\text{cm}} = \dfrac{\sum \vec{p}_i}{\sum m_i} = \dfrac{\sum m_i \vec{v}_i}{\sum m_i}$.
- 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 1 includes a quantitative and qualitative treatment of conservation of momentum in one dimension and a semiquantitative treatment of conservation of momentum in two dimensions. Exam questions involving solution of simultaneous equations are not included in AP Physics 1, but the AP Physics 1 Exam may include questions that assess whether students can set up the equations properly and reason about how changing a given mass, speed, or angle would affect other quantities. AP Physics 2 includes a full treatment of conservation of momentum in two dimensions for problems that include one unknown final velocity.

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

If the net external force on a system is zero, its total momentum is conserved 守恒:

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

Internal forces (like the push between two colliding carts) come in third-law pairs and cancel, so they cannot change the total momentum. This is the key tool for collisions 碰撞 and explosions – apply it separately in the $x$ and $y$ directions.

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

Worked example (recoil 反冲). A $60\ \text{kg}$ skater, initially at rest on frictionless ice, throws a $2.0\ \text{kg}$ ball at $8.0\ \text{m/s}$. Find her recoil speed. The total momentum starts at zero and stays zero:

$$0=(60)v+(2.0)(8.0)\;\Rightarrow\;v=-\frac{16}{60}=-0.27\ \text{m/s},$$

so she moves at $0.27\ \text{m/s}$ in the opposite direction to the ball – the principle behind rockets and guns.

A whole collection of objects can be described by a single center-of-mass velocity 质心速度:

$$\vec{v}_{\text{cm}}=\frac{\sum m_i\vec{v}_i}{\sum m_i}=\frac{\vec{p}_{\text{total}}}{M_{\text{total}}}.$$

Since total momentum is conserved when no net external force acts, $\vec{v}_{\text{cm}}$ then stays constant – a $v_{\text{cm}}$-versus-time graph is flat unless an outside force acts. An internal collision or explosion never changes it, however violently the parts fly apart.

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
collisions	碰撞	pèng zhuàng
center-of-mass velocity	质心速度	zhì xīn sù dù
recoil	反冲	fǎn chōng

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

Momentum is conserved in every collision (with no external force). Kinetic energy is not:

A glancing collision, resolved along two perpendicular axes

In an elastic collision 弹性碰撞, kinetic energy is also conserved (objects bounce apart cleanly).
In an inelastic collision 非弹性碰撞, some kinetic energy becomes heat or deformation. In a perfectly inelastic collision the objects stick together and move with one common velocity afterward.

Strategy: always write momentum conservation; add energy conservation only if the collision is stated to be elastic. One elastic fact worth memorising: in a 1D elastic collision between equal masses, the two objects simply swap velocities (a moving ball striking an identical stationary one stops dead, and the target flies off at the incoming speed).

Worked example. A $1000\ \text{kg}$ car moving at $20\ \text{m/s}$ runs into a stationary $1500\ \text{kg}$ car and they lock together. Find their common speed, and the kinetic energy lost. Momentum conservation gives

$$1000\times20=(1000+1500)\,v\;\Rightarrow\;v=\frac{20000}{2500}=8.0\ \text{m/s}.$$

Kinetic energy before is $\tfrac12(1000)(20^2)=2.0\times10^{5}\ \text{J}$; after is $\tfrac12(2500)(8.0^2)=8.0\times10^{4}\ \text{J}$. So $1.2\times10^{5}\ \text{J}$ (about $60\%$) is lost to crumpling and heat – momentum is still conserved, but kinetic energy is not.

A Newton's cradle: five steel balls hanging in a row A Newton's cradle shows momentum and kinetic energy passing through a line of balls in 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
elastic collision	弹性碰撞	tán xìng pèng zhuàng
inelastic collision	非弹性碰撞	fēi tán xìng pèng zhuàng

Exercise sheet

4.4

Exam tips

Momentum is a vector — assign $+$/$-$ signs before adding; a ball that rebounds reverses its velocity, giving a large $\Delta p$.
Momentum is conserved in every collision (no external force); kinetic energy is conserved only if the collision is stated to be elastic.
In a perfectly inelastic collision the objects stick and move with one common velocity.
Impulse $=F\,\Delta t=\Delta p$ = the area under a force–time graph; spreading a collision over a longer time reduces the force (airbags, bending knees).
For recoil/explosions, set the total momentum equal before and after (often zero before).

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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