sunrise
\(\mathbb{S}\)hereen's \(\mathcal{W}\)ebpage


I'm finally learning physics properly this summer, starting with U0 Physics (to prepare for classical mechanics in the Fall). I'm summarizing it here to keep track of things. This will be very boring for you if you have already taken U0 Physics (I'm working through the ``University Physics'' textbook), so please ignore and please don't laugh, I haven't taken a physics class since High School :)

Chapter 2: Motion Along a Straight Line
2.1: One-Dimensional Motion with Constant Acceleration
Kinematics
is the study of motion. Let \( x \) denote the position of a particle, \( x_0 \) denote the initial position, \(a \) denote the acceleration, \(v \) denote the velocity, and \(v_0\) denote the initial velocity. Then, we have the following
Kinematic Equations
, which describe the motion of a particle if we assume that the acceleration \( a \) is constant in time: \begin{align} v & = v_0 + a t \\ x & = x_0 + v_0 t + \frac{1}{2} at^2 \\ v^2 & = v_0^2 + 2a(x-x_0) \\ x-x_0 & = \frac{1}{2}(v_0 - v)t \end{align}
2.2: Freely-Falling Bodies
Free fall
is an idealized motion where we ignore the affects of (a) air resistance and (b) the curvature of the Earth. Thus, it's only suitable when the distance of the fall is small compared with the radius of the Earth (which allows us to assume a roughly constant acceleration, and apply the equations from 2.1). The only modification we make here is that we fix \( a = 9.80 m/s^2 \), which is the value for the acceleration due to gravity of a freely-falling body on the surface of the Earth.
2.3: Velocity and Position by Integrating
How can we get equations describing the motion of a particle moving in a straight line if acceleration is not constant; i.e., we're given an acceleration function \( a(t) \)? Let \( t_1 = 0 \) and let \( t_2 \) be some later time. We can obtain values for \( v(t) \) and \( x(t) \) for non-constant acceleration by integrating: \begin{align} v(t) & = v_0 + \int_0^t a(s) ds \\ x(t) & = x_0 + \int_0^t v(s) ds \end{align}
Chapter 3: Motion in Two or Three Dimensions
3.1: Position and Velocity Vectors
For notation, we denote the
position vector
by: \begin{align} \mathbf{r} := x \hat{\mathbf{i}} + y \mathbf{\hat{j}} + z \mathbf{\hat{k}} \end{align} where \(\hat{\mathbf{i}} \), \(\hat{\mathbf{j}} \), and \(\hat{\mathbf{k}} \) are the standard basis vectors of \( \mathbb{R}^3 \)
3.2: The Acceleration Vector
The
acceleration
will describe changes in the velocity magnitude (speed) and changes in the direction of velocity. the acceleration vector \( \mathbf{a}(t) \) will be tangent to the path (and thus tangent to the velocity) \( \iff \) the particle is moving in a straight line.
We can resolve the acceleration vector into its parallel (\(a_{\parallel} \)) and perpendicular (\( a_{\perp}\)) components. The parallel component tells us about changes in the speed. The perpendicular component tells us about changes in the particle's direction of motion. In its fullest generality, \begin{align} a = a_{\parallel} + a_{\perp} \end{align}
3.3: Projectile Motion
A
projectile
is a body that is given an initial velocity and then follows a path entirely determined by the affects of gravitational acceleration \( g \) and air resistance. We call the path followed by a projectile a
trajectory
.
We study an idealized system where the projectile is represented as a particle with a constant acceleration due to gravity and where the effects of air resistance, curvature, and the rotation of the Earth are all neglected (for now). We specify an initial velocity by providing the initial magnitude (i.e., speed) [denoted by \( v_0 \)] and initial direction (denoted by \( \alpha_0 \)), which is the angle that the velocity vector makes with the \(x\)-axis. The equations of motion are hence given by: \begin{align} x(t) & = (v_{0} \cos(\alpha_0))t \\ y(t) & = (v_{0} \sin(\alpha_0))t - \frac{1}{2}gt^2 \\ v_x(t) &= v_0 \cos(\alpha_0) \\ v_y(t) &= v_0 \sin(\alpha_0) - gt \end{align} From the above set of equations, we can obtain the following pieces of information: Moreover, we can derive an equation for the shape of the trajectory in terms of \(x\) and \(y\) by eliminating \(t \). We'll obtain an expression of the form \( y = bx - cx^2 \) (i.e., a parabola). The expression is: \begin{align} y = ( \tan(\alpha_0))x - \frac{g}{2v_0^2 \cos (\alpha_0)} x^2 \end{align}
3.4: Motion in a Circle
3.4.1: Uniform Circular Motion
Uniform circular motion
is when a particle moves in a circle with a constant speed. Hence, the acceleration vector will be perpendicular to the path, and also directed inward. The magnitude of the acceleration, \( a_{\text{rad}} \), is given by: \begin{align} a_{\text{rad}} = \frac{v^2}{R} \end{align} We can also call this acceleration due to uniform circular motion
centriptal acceleration
. We define the
period
of a motion to the time it takes for one revolution to occur. We can re-write the centripital acceleration in terms of the period: \begin{align} a_{\text{rad}} = \frac{4 \pi^2 R}{T^2} \end{align}
3.4.2: Non-uniform circular motion
If the speed varies, then acceleration will also have a tangential component: \begin{align} \mathbf{a} = a_{\text{rad}} + a_{\text{tan}} = \frac{d || \mathbf{v} ||}{dt} + \frac{v^2}{R} \end{align}
3.5: Relative Velocity
The
relative velocity
is the velocity of a particular observer relative to that observer.
3.5.1: Relative Velocity in One Direction
A
Frame of Reference
is an observer equipped with a meter-stick and stop-watch. Notation: let \( P \) be a point, and let \( A \) and \( B \) be reference frames. Then, \( x_{P/A} \) will denote the position of \( P \) relative to the reference frame \( A \). Moreover, the position of the origin of \( B \) with respect to the origin of \( A \) is \( x_{B/A} \). We thus have: \begin{align} x_{P / A} & = x_{P / B} + x_{B/A} \\ v_{P/A} & = v_{P / B} + v_{B/A} \end{align}
3.5.2: Relative Velocity in Two or Three Dimensions
It's exactly what we would expect. Let \( \mathbf{r} \) denote the position vector. Then: \begin{align} \mathbf{r}_{P/A} & = \mathbf{r}_{P/B} + \mathbf{r}_{B/A} \\ \mathbf{v}_{P/A} & = \mathbf{v}_{P/B} + \mathbf{v}_{B/A} \end{align} We call the above transformation the
Galilean Velocity Transformation
. We also have the general rule that for all reference frames \( A \) and \( B \), \begin{align} \mathbf{v}_{A/B} = - \mathbf{v}_{B / A} \end{align}
Chapter 4: Newton's Laws of Motion
Dynamics
is the study of the relationships between motion and the forces which cause motion. The basis of this topic are
Newton's Laws of Motion
, and this forms the foundation of
classical mechanics
, which is a very good description of reality so long as we are not working with relativistic speeds (speeds comparable to the speed of light) or small objects (atomic scale).
4.1: Force and Interactions
A
force
is an interaction between a body and its surroundings or between two objects. It's a vector quantity. Types of forces:
4.1.1: Superposition of Forces
The principle of the
superposition of forces
states that any number of forces applied to a body have the same affect as a single force that is the sum of all the individual forces. This is called the
net force
acting on a body. We denote it by: \begin{align} \mathbf{R} = \sum_{i=1}^N \mathbf{F}_i \end{align} and we can find the magnitude and direction of \( \mathbf{R} \): \begin{align} || \mathbf{R} || = \sqrt{R_x^2 + R_y^2} \hspace{1cm} \tan (\theta) = \frac{R_y}{R_x} \end{align}
4.2: Newton's First Law
Newton's First Law of Motion: A body acted upon by no net force has a constant velocity and hence zero acceleration. The tendency of a body to either remain at rest or keep moving once its set in motion is called inertia. We say that an object is in equilibrium if an object is either at rest or moving with a constant velocity. Hence, Newton's First Law states that for an object in equilibrium, one has: \begin{align} \sum_{i=1}^N \mathbf{F}_i = \mathbf{0} \end{align} where \( \mathbf{F}_1 \), ... , \( \mathbf{F}_N \) are the forces acting on the body.
4.2.1: Inertial Frames of Reference
An
inertial reference frame
is a reference frame where Newton's First Law is valid. The Earth is approximately an inertial reference frame. Given one inertial reference frame \( A \), we can construct any other reference frame \( B \) by considering a reference frame that is moving with a constant velocity relative to \( A \).
4.3: Newton's Second Law
Many experiments show that for a given body, the ratio of the magnitude of the net force, \( || \sum_{i=1}^N \mathbf{F}_i || \), to the magnitude of the acceleration, \( || \mathbf{a} || \), is constant. Hence, we have a quantitative measure of inertia: \begin{align} m = \frac{|| \sum_{i=1}^N \mathbf{F}_i ||}{|| \mathbf{a} ||} \end{align} We define this ratio to be the
inertial mass
. We can thus express Newtons (the unit of force) in terms of the standard units as follows: \begin{align} 1 N = 1 \text{kg} \cdot m/s^2 \end{align} Experiments also verify that what matters in terms of acceleration is the net force; these forces are what cause objects to accelerate.
Newton's Second Law of Motion: A body accelerates if a non-zero net force acts on the body. This acceleration is in the same direction as the net force. Mathematically, \begin{align} \sum_{i=1}^N \mathbf{F}_i = m \mathbf{a} \end{align}
Four remarks on Newton's Second Law:
  1. It's a vector equation, and so to analyze we break it down into component-form.
  2. It only includes external forces
  3. \(m \) must be constant; for systems where the mass varies, momentum is a more suitable concept.
  4. As with Newton's First Law, it's only valid in inertial reference frames
4.4: Mass and Weight
The
weight
of an object is the gravitational force exerted on a body by the Earth. Hence, \begin{align} \mathbf{w} = m \mathbf{g} \end{align}
4.5: Newton's Third Law
Newton's Third Law of Motion: forces come in two pairs; if a body \( A \) exerts a force on a body \( B \), then \( B \) exerts a force on \(A \) that is equal in magnitude but in the opposite direction. Mathematically, \begin{align} \mathbf{F}_{\text{ \(A \) on \( B \) }} = - \mathbf{F}_{\text{ \( B \) on \( A \) }} \end{align}
We call the pair \( \mathbf{F}_{\text{ \(A \) on \( B \) }} \) and \( \mathbf{F}_{\text{ \( B \) on \( A \) }} \) an
action-reaction pair
. Two forces in an action-reaction pair never act on the same body.
4.6: Free-body Diagrams
Free-body
diagrams are an important tool used to solve physics problems. Some things to keep in mind when solving physics problems:
  1. Newton's first and second laws can only be applied once you choose the that you are going to analyze.
  2. We only consider the forces acting on the body.
  3. If there are multiple bodies, then you need to take the problem apart and draw a free-body diagram for each body.
Chapter 5: Applying Newton's Laws
5.3: Friction Forces
When a body is on a surface, that surface exerts a single contact force on that body. We can resolve this force into its parallel and perpendicular components. The perpendicular component is the normal force \( \mathbf{n} \). The parallel component is the
friction force
, denoted by \( \mathbf{f} \). The direction of the friction force is always opposite to the direction of the relative motion.

There are two types of friction force:
  1. Kinetic friction force
    : denoted \( \mathbf{f}_k \), this is friction that acts when an object is moving along a surface. It has been experimentally verified that:
  2. \begin{align} || \mathbf{f_k} || = \mu_k || \mathbf{n} || \end{align} The
    coefficient of kinetic friction
    , denoted \( \mu_k \), is the proportionality constant between the forces. The slippier the surface, the smaller \( \mu_k \) is. It's a dimensionless quantity.
  3. Static friction force
    , denoted \(\mathbf{f}_s \), is the friction that acts on an object that is not moving relative to the surface. The maximum value that the static friction force can attain before the object starts moving -- \( ( \mathbf{f}_s )_{\text{max}} \) is proportional to \( || \mathbf{n} || \). \( \mu_s \) is the proportionality factor: \begin{align} || \mathbf{f}_s || \leq ( \mathbf{f}_s )_{\text{max}} = \mu_s || \mathbf{n} || \end{align} and we call it the
    Coefficient of static friction
    .
The coefficient of static friction is usually greater than the coefficient of kinetic friction.
5.3.1: Rolling Friction
The
coefficient of rolling friction
, denoted \( \mu_T \), tells us how much easier it is to move something on wheels, as opposted to sliding it. It is given by: \begin{align} \mu_T = \frac{\text{horizontal force needed for a constant speed on a flat surface}}{\text{upward normal force exerted by the surface}} \end{align}
5.3.2: Fluid Reistance and Terminal Speed
Fluid resistance
is a force exerted by a fluid on a body which is moving through the fluid. The direction of the fluid resistance force is always opposite the direction of the body's velocity with respect to the fluid. The existence of fluid resistance means that objects falling through a fluid do not have constant acceleration, and hence the kinematic equations that we've been using all along are no longer valid. We observe that as speed increases, the resisting force increases until \( mg - kv_y = 0 \). At this point, acceleration becomes zero, and velocity no longer increases, and we attain what is called the
terminal speed
. We denote this by \(v_t \), and it is given by: \begin{align} \mathbf{v}_t = \frac{m \mathbf{g}}{k} \end{align} We can obtain expressions for \( \mathbf{v}_y \), \( \mathbf{a}_y \), and \( y \) using Newton's second law: \begin{align} m \frac{d \mathbf{v}_y}{dt} = m \mathbf{g} - k \mathbf{v}_y \end{align} This is a differential equation in \( v_y \); solving for this, and then differentiating for acceleration and integrating for position, we obtain: \begin{align} \mathbf{v}_y & = \mathbf{v}_t \left[ 1 - e^{-(k/m)t} \right] \\ \mathbf{a}_y & = \mathbf{g} e^{(-k/m)t} \\ \mathbf{y} & = \mathbf{v}_t \left[ t - \frac{m}{k} \left( 1 - e^{-(k/m)t} \right) \right] \end{align} The terminal velocity in the case of \( || \mathbf{f} || = D || \mathbf{v} ||^2 \) is: \begin{align} || \mathbf{v}_t || = \sqrt{ \frac{m || \mathbf{g} ||}{D}} \end{align}
5.4: Dynamics of Circular Rotation
Recall that the magnitude of the acceleration of an object in unoform circular motion is given by: \begin{align} a_{\text{rad}} = \frac{|| \mathbf{v} ||^2}{R} \end{align} or, equivalently, in terms of the period \( T \): \begin{align} a_{\text{rad}} = \frac{4 \pi^2 R}{T^2} \end{align} These forces and accelerations are governed by Newton's Second Law, since the particle is accelerating towards the centre, and hence \( \sum \mathbf{F} \) must be directed inwards. If this force is zero, then the particle will fly off in the tangent direction. In uniform circular motion, the magnitude \( || \sum \mathbf{F} || \) is constant since \( || \mathbf{a} || \) is constant. The magnitude of the force \( \mathbf{F}_{\text{net}} \) is hence: \begin{align} || \mathbf{F}_{\text{net}} || = m a_{\text{rad}} = m \frac{|| \mathbf{v} ||^2}{R} \end{align} Remark: these expressions are valid for any circular arc; we don't require the particle to complete a full circle.
5.5: The Fundamental Forces of Nature
  1. Graviational Interactions
    : e.g. weight, planets in orbit \( \rightarrow \) Ch. 13
  2. Electromagnetic Interaction
    : electric and magnetic fores, e.g. contact forces (e.g. normal force, friction, fluid reistance).
  3. Strong Interaction
    : holds the nucleus of an atom together.
  4. Weak Interaction
    : responsible for beta decay.
In the 1960s, the electromagnetic and weak interactions were unified into an ``
electroweak interaction
.''
Chapter 6: Work and Energy
6.1: Work
The total work done on a particle by all forces acting on the object will tell us the change in the particle's kinetic energy, a quantitty which is related to the particle's mass and speed. This rule even holds when the force varies.
Work I: Consider a particle which undergoes a constant displacement, \( || \mathbf{s} || \), in a straight line. Let \( \mathbf{F} \) be the constant force acting upon the particle, parallel to the displacement vector. Then, the work \( W \) done on the particle is the product: \begin{align} W := || \mathbf{F} || || \mathbf{s} || \end{align} We can generalize this to a displacement vector not necessarily parallel to the force vector; in this case, let \( \varphi \) be the angle between \( \mathbf{F} \) and \( \mathbf{s} \). Then, the work is: \begin{align} W = || \mathbf{F} || || \mathbf{s} || \cos (\varphi) \end{align} This is the dot product: \begin{align} W = \mathbf{F} \cdot \mathbf{s} \end{align}
The SI unit for work is the joule \( \sim \) \( N \cdot m \).
6.2: Kinetic Energy and the Work-Energy Theorem
The
kinetic energy
of a particle with mass \( m \) and velocity \( \mathbf{v} \) is given by: \begin{align} K = \frac{1}{2} m || \mathbf{v} ||^2 \end{align} One can use Newton's Second Law + Kinematic Eqns to derive:
Work-Energy Theorem: The work done by the net force on a particle is given by the change in kinetic energy: \begin{align} W_{\text{tot}} = \Delta K \end{align}
Since this theorem is derived using Newton's Second law, it is only valid in intertial reference frames. The Work-Energy theorem is useful when one wants to relate a body's speed \( || \mathbf{v}_1 || \) at one point in the trajectory to its speed \( || \mathbf{v}_2 || \) at another point in its trajectory.
6.2.1: The Meaning of Kinetic Energy
The physical interpretation of kinetic energy or a particle is that it's the work required to bring that particle from rest to that speed. An equivalent way to look at kinetic energy is to see it as the work a particle can do as it's brought to rest fro a certain speed.
6.3: Work and Energy with Varying Forces
Work II: the work done on a particle with a varying x-component of force, \( F_x \), along a straight-line displacement along the x-axis is: \begin{align} W = \int_{x_1}^{x_2} F_x dx \end{align}
Application: consider a spring which is stretched by an amount \( x \). Then, the force required to stretch the spring is: \begin{align} F_x = k x \end{align} where \( k \) is called the
force constant
, \( k \sim N/m \). This observation is known as
Hooke's Law
.
6.3.1: Work-Energy Theorem for Motion Along a Curve
We can now generalize the definition to include a force that varies in both magnitude and direction; here, work is defined as a line integral:
Work III: the work done on a particle by a varying force \( \mathbf{F} \) along a curved path is given by: \begin{align} W = \int_{P_1}^{P_2} \mathbf{F} \cdot \mathbf{d \ell} = \int_{P_1}^{P_2} || \mathbf{F} || \cos (\varphi) d \ell = \int_{P_1}^{P_2} F_{\parallel} d \ell \end{align} where \( \varphi \) is the angle between \( \mathbf{F} \) and \( \mathbf{d \ell} \).
6.4: Power
We need a measure of how quickly work is done \( \Rightarrow \) power. The
Power
is the time rate at which work is done: \begin{align} P := \frac{dW}{dt} \end{align} The SI unit is the Watt, denoted \( W \); \( W \sim J/s \). We also have another definition of power: \begin{align} P := \mathbf{F} \cdot \mathbf{v} \end{align}
Chapter 7: Potential Energy and Energy Conservation
7.1: Gravitational Potential Energy
Potential energy
: the potential energy associated with the position of a particle. One particular type is
Gravitational potential energy
, which is the potential energy associated with the body's weight and height above the ground. It is given by: \begin{align} U_{\text{grav}} := mgy \end{align} The work done by the gravitational force on a particle is: \begin{align} W = - \Delta U_{\text{grav}} \end{align}
7.1.1: Conservation of Mechanical Energy (Gravitational Forces Only)
If only the gravitational force does the work, then the total mechanical energy is conserved: \begin{align} K_1 + U_{\text{grav, 1}} = K_2 + U_{\text{grav, 2}} \end{align} We call the sum \( E := K + U_{\text{grav}} \) the
total mechanical energy
of the system. Hence, when only gravity is doing the work, \begin{align} E = K + U_{\text{grav}} = c \end{align} for \( c \in \mathbb{R} \). Mechanical energy is conserved; this principle is called the
conservation of mechanical energy
.
7.1.2: When Forces Other Than Gravity Do Work
Let \( W_{\text{other}} \) be the work done by the force \( \mathbf{F}_{\text{other}} \), forces which are not the gravitational force. Then, one has: \begin{align} K_1 + U_{\text{grav, 1}} + W_{\text{other}} = K_2 + U_{\text{grav, 2}} \end{align} That is, \( W_{\text{other}} \) is the change in the system's total mechanical energy, \( E = K + U_{\text{grav}} \).
7.2: Elastic Potential Energy
The
elastic potential energy
is the energy stored in a deformable body such as a string. An object which returns to its original size and shape after being defored is said to be
elastic
. Elastic potential energy is given by: \begin{align} U_{\text{el}} := \frac{1}{2} kx^2 \end{align} where \( U_{\text{el}} \) denotes the elastic potential energy stored in a spring, \( k \) is the force constant, and \( x \) is the displacement from the spring's natural length. The work done by the elastic force is given by: \begin{align} W_{\text{el}} = U_{\text{el}, 1} - U_{\text{el}, 2} = - \Delta U_{\text{el}} \end{align} From the Work-Energy theorem, the total mechanical energy is conserved if only the elastic force is doing the work: \begin{align} K_1 + U_{\text{el}, 1} = K_2 + U_{\text{el}, 2} \end{align}
7.2.1: Situations with Both Gravitational and Elastic Potential Energy
Most general statement: the work fone on a system by all other forces, i.e., the non-gravitational and non-elastic forces, is the change in the system's total mechanical energy: \begin{align} K_1 + U_1 + W_{\text{other}} = K_2 + U_2 \end{align} where here, \( K_1 \) is the initial kinetic energy, \( U_1 \) is the initial potential energy of all kinds, \( W_{\text{other}} \) is the work done by other non-potential energy forces, \( K_2 \) is the final kinetic energy, and \( U_2 \) is the final potential energy of all kinds.
7.3: Conservative and Non-Conservative Forces
A
conservative force
is a force where the sum of the kinetic and potential energy remains the same for all time. Exampels of conservative forces include gravity and the spring force. The work done by conservative forces obeys the following four properties:
  1. Work can be expressed as the difference between initial and final values of a
    potential-energy
    function.
  2. Reversible
  3. Path-independence: work does not depend on the path the body takes.
  4. When \( x_1 = x_2 \), the total work done is zero.
Mechanical energy is conserved if the only forces doing the work are conservative forces.
7.3.1: Non-Conservative Forces
A
non-conservative force
is a force that is not conservative; it cannot be represented by a potential-energy function. A sub-class of non-conservative forces are
dissipative forces
, which are non-conservative forces which cause mechanical energy to be lost or dissipated. An example of this is kinetic friction or fluid resistance.
7.3.2: Law of Conservation of Energy
We can represent non-conservative forces in terms of
internal energy
, which is the energy associated with the change in the state of the materials. For example, raising the temperature of an object raises its internal energy. The following has been experimentally verified:
Law of Conservation of Energy: The sum of all types of energies remains the same. Energy is never created nor destroyed, but it can change form: \begin{align} \Delta K + \Delta U + \Delta U_{\text{int}} = 0 \end{align} where \( \Delta K \) is the change in the kinetic energy, \( \Delta U \) is the change in the potential energy, and \( \Delta U_{\text{int}} \) is the change in the internal energy.
The branch of physics that studies the relationship between internal energy and temperature changes, heat, and work is called thermodynamics.
7.4: Force and Potential Energy
We have the following expression which gives us the force corresponding to a given potential-energy expression, of course, assuming that the force is a conservative force. Let \( F_x (x) \) be the x-component of the force and let \( U(x) \) be the associated potential energy function. Using the formula for work and taking the limit, we obtain that: \begin{align} F_x(x) = - \frac{ dU(x)}{dx} \end{align} The physical meaning behind this is that conservative forces always act to push the system towards the lowest potential energy.
7.4.1: Force and Potential Energy in Three Dimensions
\begin{align} \mathbf{F} = - \left( \frac{\partial U}{\partial x} \mathbf{\hat{i}} + \frac{\partial U}{\partial y} \mathbf{\hat{j}} + \frac{\partial U}{\partial z} \mathbf{\hat{k}} \right) = - \nabla \mathbf{U} \end{align}
7.5: Energy Diagrams
An
energy diagram
is a graph that shows the potential energy function \( U(x) \) and the energy of a particle \( E \) of a particle subjected to a force with a corresponding potential energy function \( U(x) \).
  1. Stable equilibrium
    : a minimum of a potential-energy curve.
  2. Unstable equilibrium
    : a maximum of a potential-energy curve.
Chapter 8: Momentum, Impulse, and Collisions
8.0: Introduction and Motivation
A key part of Newtonian mechanics is the conservation of momentum; this allows us to study situations that may be very difficult to analyze with Newton's Laws. In particular, the conservation of momentum is valid in situations where Newton's Laws are not valid. We use this principle when studying collision problems and problems with changing masses.
8.1: Momentum and Impulse
We'll see another useful way to re-state \( \mathbf{F} = m \mathbf{a} \), as we did by previously re-stating it as the Work-Energy theorem.
8.1.1: Newton's Second Law in terms of Momentum
We can write Newton's second law as, \begin{align} \sum \mathbf{F}_i = \frac{d}{dt} [ m \mathbf{v} ] \end{align} The
momentum
or
linear momentum
of a particle is defined as: \begin{align} \mathbf{p} = m \mathbf{v} \end{align} It's SI unit is \( kg \times m/s \). We can re-formulate Newton's Second Law in terms of momentum: \begin{align} \sum \mathbf{F}_i = \frac{d \mathbf{p}}{dt} \end{align} This is only valid in intertial reference frames.
8.1.2: Impulse-Momentum Theorem
Question: What is the physical difference between momentum and kinetic energy? We can answer this question by introducing the concept of impulse. The
impulse
of a constant net force \( \sum \mathbf{F}_i \) during some time interval \( \Delta t := [t_1, t_2] \) is defined as: \begin{align} \mathbf{J} := \sum \mathbf{F} ( t_2 - t_1) = \sum \mathbf{F} \Delta t \end{align} Its SI unit is \( N \cdot s \sim kg \times m/s \).
Impulse-Momentum Theorem (Constant Forces): The impulse of a net force on a particle during a time interval is given by: \begin{align} \mathbf{J} = \mathbf{p_1} - \mathbf{p_2} = \Delta \mathbf{p} \end{align} For non-constant forces, we have: \begin{align} \mathbf{J} = \int_{t_1}^{t_2} \sum \mathbf{F}_i dt \end{align}
We can also use the average net force: \begin{align} \mathbf{J} = \mathbf{F}_{\text{avg}} (t_2 - t_1) \end{align}
8.1.3: Momentum and Kinetic Energy Compared
On one hand, we have the impulse-momentum theorem, \begin{align} \mathbf{J} = \mathbf{p}_1 - \mathbf{p}_2 \end{align} which gives us the changes in a particle's momentum due to impulse. This depends on the time over which the net force acts. On the other hand, we have the work-energy theorem, \begin{align} W_{\text{tot}} = K_2 - K_1 \end{align} which gives us changes in the particle's kinetic energy due to work. This depends on the distance over which the net force acts. The physical meaning behind all of this is: The impulse-momentum theorem and the work-energy theorem are examples of integral principles, principles which relate the motion of two different times separated by some finite time-interval. Conversely, Newton's Second Law is an example of a differential principle, which relates the rate of change of the velocity or momentum to the force.
8.2: Conservation of Momentum
Internal Forces
are forces that the particles in a system exert on each other.
External Forces
are forces exerted on any part of the system by some object outside of it. An
Isolated System
is one where there are no external forces. Denote by \( \mathbf{F}_{\text{B on A}} \) the net force on a particle \( A \) and denote by \( \mathbf{F}_{\text{A on B}} \) the net force on a particle \( B \). Then, for an isolated system, we have \begin{align} \mathbf{F}_{\text{B on A}} = \frac{d \mathbf{p}_A}{dt} \hspace{1cm} \mathbf{F}_{\text{A on B}} = \frac{d \mathbf{p}_B}{dt} \end{align} By Newton's Third law, \begin{align} \mathbf{F}_{\text{B on A}} + \mathbf{F}_{\text{A on B}} = \frac{d \mathbf{p}_A}{dt} + \frac{d \mathbf{p}_B}{dt} = \frac{d(\mathbf{p}_A + \mathbf{p}_B)}{dt} \end{align} We define the
Total Momentum
, \( \mathbf{P} \), as the vector sum of the momenta of the individual particles of the system. Hence, the above formula is saying that the total momenta of an isolated system is constant. This gives us the first form of the Principle of the Conservation of Momentum:
Principle of Conservation of Momentum I: If the vector sum of the external forces on a system is zero, then the total momentum of a system, \( \mathbf{P} \), is constant.
8.3: Momentum, Conservation, and Collisions
8.3.1: Elastic and Inelastic Collisions
  1. Elastic Collision
    : when the forces acting on the bodies are conservative forces, then by the conservation of mechanical energy, no mechanical energy is lost, and hence kinetic energy remains the same.
  2. Inelastic Collision
    : collision in which the total kinetic energy after the collision is less than the collision
    1. A sub-class:
      Completely Inelastic Collision
      : inelastic collision where the bodies collide into one body and then move together.
Note. For any collision where the external forces are negligible, then the total momentum is conserved; for any collision that is elastic, the kinetic energy is conserved.
8.3.2: Completely Inelastic Collision
One can use the conservation of momentum to prove that kinetic energy is always lost in a completely inelastic collision.
8.3.3: Classifying Collisions
  1. Kinetic energy conserved \( \rightarrow \) elastic.
  2. Kinetic energy decreases \( \rightarrow \) inelastic.
  3. Two bodies have a common final velocity \( \rightarrow \) completely inelastic.
8.4: Elastic Collisions
Elastic collisions occur when the forces between colliding bodies is conservative. For elastic collisions, we have also the conservation of kinetic energy. This property, plus the conservation of momentum, allows us to find the final velocityes \( v_{A2} \) and \( v_{B2} \). Let's consider the special case when \( v_{B1} = 0 \) (for straight-line motion). Then, \begin{align} v_{A2} & = \frac{m_A - m_B}{m_A + m_B} v_{A1} \\ v_{B2} & = \frac{2m_A}{m_A + m_B} v_{A1} \end{align}
8.4.1: Elastic Collisions and Relative Velocity
In a straight-line elastic collision of two bodies, the relative velocities before and after the collision have the same magnitude but the opposite sign. When this condition is satisfied, the total kinetic energy is also conserved.
8.5: Centre of Mass
Suppose that we have several particles with masses \( m_1, m_2, ... \) with position vectors \( \mathbf{r}_1, \mathbf{r}_2, ... \). Then, the
centre of mass
of the system of particles is given by \begin{align} \mathbf{r}_{cm} := \frac{\sum_{i} m_i \mathbf{r}_i}{\sum_{i} m_i} \end{align} We call this the mass-weighted average position of the particles.
8.5.1: Motion of the Centre of Mass
Again, let \( m_1, m_2, ... \) denote the masses of the particles of the system. Let \( M \) denote the total mass of the system, \begin{align} M := m_1 + m_2 + m_3 + ... \end{align} Then, the
velocity of the centre of mass
is given by \begin{align} \mathbf{v}_{cm} := \frac{m_1 \mathbf{v}_1 + m_2 \mathbf{v}_2}{m_1 + m_2+ ...} \end{align} We can re-write this to obtain the total momentum of the system, \begin{align} M \mathbf{v}_{cm} = m_1 \mathbf{v}_1 + m_2 \mathbf{v}_2 + ... = P \end{align} Hence, the total momentum \( P \) of a system equals the total mass times the velocity of the centre of mass. Observe: for a system of particles with zero net force, then we know by Newton's Second Law that this implies that \( P \) is constant. By the above, this implies that the velocity of the centre of mass is constant.