When you throw a ball, kick a soccer ball, or watch a basketball arc towards the hoop, you’re witnessing a classic parabolic trajectory. We learn in introductory physics that this path is the result of constant horizontal velocity and constant vertical acceleration due to gravity. But what if I told you that this parabola is an illusion, a small piece of a much grander, cosmic dance? What if your football, for that brief moment it’s in the air, is actually on a suborbital flight, tracing a tiny segment of an enormous ellipse that stretches to the center of the Earth?

This isn’t just a matter of perspective; it’s a profound connection between the everyday physics of projectiles and the celestial mechanics that govern the orbits of planets and moons. The key to this connection lies in a single, powerful concept: gravity.

Key takeaway

• Parabolic Motion is an Illusion Assuming Constant Gravity: The familiar parabolic path of a thrown object is just a small, visible segment of a much larger elliptical orbit with the assumption that gravity is constant in direction and magnitude.
• Inverse-Square Law of Gravity Yields Elliptical Trajectories: We see a parabola because we assume gravity is constant in the short trajectory above ground (always pulling straight down with the same force). In reality, gravity is an inverse-square force that always pulls toward the Earth’s center, which decreases as the object moves farther away. This mathematically results in an elliptical path, like any satellites do.
• A Cosmic Perspective of Local Parabola: For its brief flight, a thrown ball is a suborbital satellite. Its full elliptical path has one focus at the center of the Earth, and its highest point is the far end of the ellipse’s long axis. The only reason it falls back to Earth is that this skinny elliptical path intersects the ground.
• Math Putting Everything Together: With a little math, we will show that the tiny small section at the end of a very elongated ellipse locally approximates to a parabola, just like what you would expect from a constant gravitational field.

The Two Views of Gravity

Now back to the long view. The reason we perceive the path of a thrown object as a parabola is that we make a simplifying assumption: that gravity is constant. We assume that the force of gravity pulls the object straight down with the same strength, regardless of its position. For the relatively short distances a football travels, this approximation works remarkably well.

However, the reality, as described by Newton’s Law of Universal Gravitation, is that gravity is not constant. It’s a force that follows an inverse square law, meaning it weakens with the square of the distance between two objects, and its direction always points toward the center of the Earth. This is the same principle that governs the interactions between planets and the Sun, or a satellite and the Earth. When you solve the equations of motion with this more accurate model of gravity, the resulting trajectory is not a parabola, but an ellipse (or a hyperbola or parabola if the object is moving fast enough to escape).

So, our football doesn’t have nearly enough speed to complete a full orbit and become a new moon, so it inevitably succumbs to gravity and falls back to Earth. But for its brief flight, it is, in a very real sense, a satellite on a suborbital path. The parabolic trajectory we see everyday is a local approximation of a much larger elliptical orbit that holds true when gravity can be seen as constant in both magnitude and direction.

Visualizing the Full Orbit

So what does this full elliptical orbit of your thrown ball look like? It’s a very long and skinny ellipse. One of the two focal points of this ellipse is at the center of the Earth. The other focal point is very close to the Earth’s surface.

For a short throw, the semi-major axis is roughly the radius of the Earth, while the semi-minor axis is much, much smaller. Because the semi-minor axis is so small, this elliptical path inevitably intersects with the surface of the Earth, which is why the object falls back down. For an object to complete a full orbit without crashing, a necessary condition is for both its semi-major and semi-minor axes to be longer than the Earth’s radius, ensuring its path never crosses the ground.

The highest point of the ball’s trajectory (the apex of the “parabola”) is actually one end of the major axis (the long axis) of this grand ellipse.

From Ellipse to Parabola: The Math

We can mathematically demonstrate how the familiar parabola emerges from the equation of an ellipse under the conditions of a typical projectile on Earth.

The standard equation for an ellipse centered at the origin is:

where a is the semi-major axis and b is the semi-minor axis.

To better represent our thrown object, let’s shift the coordinate system so that the origin is at the apex of the trajectory, which is one end of the major axis. We also switch x and y axis because we customarily use y as the vertical axis, which is the now along the major axis direction. The equation becomes:

For a projectile thrown on Earth, the trajectory is very flat, meaning the semi-major axis is much, much larger than the semi-minor axis (a ≫ b). Also, the distances we observe, x and y, are tiny compared to the semi-major axis a.

Let’s expand the first term:

$latex \begin{align} \frac{y^2 – 2ay + a^2}{a^2} + \frac{x^2}{b^2} &= 1 \
\frac{y^2}{a^2} – \frac{2y}{a} + 1 + \frac{x^2}{b^2} &= 1
\end{align}$

Now, since y is very small compared to a, we have

So the first term can be ignored. Then this leaves us with:

Rearranging for x^2, we get:

This is the equation of a parabola opening sideways. By rotating our coordinate system, we would see the familiar parabolic shape of projectile motion. This derivation beautifully illustrates that the parabolic path we observe is a direct mathematical approximation of an elliptical orbit under the conditions of “short” distances and “weak” initial velocities.

We can go one-step further to calculate the two axis of the ellipse. Here we are not going to the mathematical details, but just to give the result here. Suppose the initial speed of the throw is v₀ at angle θ relative the ground. We also know the Earth mass is M radius R. We can derive
$latex \begin{align} \text{Ratio of major-minor axis } \frac{b}{a} &= \sqrt{1-e^2} \ \text{Ellipticity } e &= \sqrt{1-\frac{h^2}{GMa}} \ \text{Angular momentum } h &= v_0R\sin{\theta} \
\text{Gravity accelation } g &= \frac{GM}{R^2} \end{align}$
Therefore, the parabolic equation becomes
$latex \begin{align} y &= \frac{a}{2b^2}x^2 \
&= \frac{1}{2a}(\frac{a}{b})^2x^2 \
&= \frac{1}{2a(1-e^2)}x^2 \
&= \frac{1}{2a (1-(1-\frac{h^2}{GMa}))}x^2 \
&= \frac{GM}{h^2}x^2 \
&= \frac{GM}{2v_0^2R^2\sin^2{\theta}}x^2 \
&= \frac{g}{2v_0^2\sin^2{\theta}}x^2
\end{align}$
This is exactly the parabolic projectile trajectory you would get from constant gravity g.

A Cosmic Perspective on a Simple Throw

So, the next time you toss a ball in the air, remember what you’re truly witnessing. You’re not just seeing a simple arc; you’re observing a tiny piece of a grand, elliptical journey. That ball is, for a fleeting moment, a celestial body on a suborbital flight, tethered to the center of the Earth by the same laws of gravity that choreograph the dance of the cosmos. It’s a humbling and awe-inspiring reminder that the universe’s most profound principles are at play in our most ordinary actions.

In Part 1 of our pyrotechnic physics journey, we marveled at how, at any given instant, the fragments of an exploding firework form a perfect, falling sphere. We can also look at a different perspective of the picture. Each of those glittering stars traces its own distinct parabolic path through the night sky. If you could see all these paths traced out simultaneously over their entire flight, there seems to be a dome or a ultimate boundary that contains them all.

This boundary is what we call the “envelope of safety” or the “parabola of safety.” It’s the curve that no projectile from that initial burst (given a fixed initial speed) can cross. Understanding this envelope is not just an academic curiosity; it has real-world implications for ensuring the safety of spectators and infrastructure during a display.

So, welcome back! In this second and concluding part, we’ll dive deep into the mathematics and physics required to define this crucial outer limit.

1. Visualizing the Challenge: The Envelope of Trajectories

a. The Intuitive Image

Imagine a time-lapse photo of a single firework shell bursting, but instead of dots of light, each fragment leaves a continuous glowing trail from the moment of explosion until it fizzles out or hits the ground. You’d see a beautiful, intricate web of overlapping parabolic arcs, all originating from the same point but fanning out in different directions.

While the individual stars form a sphere at any instant, the collection of their entire paths sketches out a different, larger shape. This overall boundary is what we’re after.

b. Conceptualizing the Problem

Our goal is to find a single curve, which we can denote as:

This curve “envelops” all possible individual projectile trajectories. Each individual trajectory can be described by y = f(x, θ), where θ is the launch angle (or some other parameter related to it) and v₀ is the initial speed. No projectile launched from the origin with this initial speed should be able to reach a point outside this envelope.

2. Mathematical Toolkit: Understanding Envelope Curves

What exactly is an envelope curve in mathematics? Consider a family of curves, where each curve in the family is distinguished by a parameter (like our launch angle θ). An envelope curve has some fascinating properties:

a. Properties of an Envelope Curve Yₑ = G(X)

Let our family of curves be described by an equation involving a parameter θ (which we’ll later relate to our launch angle):

The envelope, Yₑ = G(X), has the following characteristics:

1. Tangency: At every point on the envelope, it is tangent to one of the curves in the family y = f(x, θ). The envelope “just kisses” each curve it encloses, or is made up of these points of tangency.
2. Extremal Property: For a given x (or y), the corresponding point on the envelope often represents an extreme value (maximum or minimum) of y (or x) for the family of curves at that point. In our firework scenario, for any horizontal distance x from the burst, the envelope will give the maximum possible height Yₑ that any projectile can reach. So,
   
3. Derivative Condition: This extremal property leads to a powerful mathematical condition. If we’re looking for the maximum y for a given x by varying the parameter θ, we can find this by setting the partial derivative of f(x, θ) with respect to θ to zero. Also because all points on the envelope curve are also on at least one of the original equation for the family of curves. So all points on envelope curve need to satisfy the following system of equations:
   
   Solving this system of equations, eliminating θ, gives the equation of the envelope. This is the formal form of deriving the envelope of a family of curves y=f(x,θ).
4. Converging Intersections (Locus of Ultimate Intersections): Another intuitive understanding of the envelope curve is to see it as the convergence of the intersection of two curves differing by a small change in θ.

Two curves in the family for a given θ but differing by a small change Δ θ can be described as below: As Δ θ approaches zero, the intersection points of these neighboring curves approach a point on the envelope. The envelope is the collection of all locus of these ultimate intersection points for all starting angle θ.

These properties give us several avenues to mathematically derive the envelope of our firework trajectories.

3. Back to the Physics: The Projectile’s Path

Let’s recall the trajectory equation for a single projectile launched from the origin (0,0) with initial speed v₀ at an angle θ to the horizontal (ignoring air resistance, and with gravity g acting in the negative y-direction). Let t be the time of flight for the projectile.

1. Parametric Equations:
   The position components are given by:
   
2. Trajectory Equation y = f(x, θ):
   As we have done in Part 1, we can get the trajectory equation y = f(x, θ). To get the path in terms of x and y, we eliminate t from the x equation:
   
   Substituting this into the y equation:
   
   This simplifies to:
   
   Here we go one step further, using the following trigonometry identity
   
   So, we get the trajectory equation:
   
   This quadratic equation of y(x) describes the parabolic path of a single projectile launched at angle θ.
3. Substitution with m = \tan θ:
   To simplify differentiation or algebraic manipulation, let the parameter m be:
   
   The parameter m can range from −∞ to +∞, covering all possible launch angles (except directly up/down, which are limiting cases). Our family of curves becomes y = f(x, m):
   
   This is the equation we will work with to find the envelope, with m as our parameter.

4. Finding the Parabola of Safety: Multiple Paths to One Truth

Now, let’s employ the different mathematical strategies suggested by the properties of envelope curves.

a. The Extrema Method: Maximizing Height for Each x (by Completing the Square)

For any given horizontal distance x from the launch point, we want to find the launch parameter m = \tan θ that maximizes the height y. This maximum y will be a point on the envelope. Our trajectory equation is:

Let’s rearrange this into a standard quadratic form in terms of m:

For a fixed x, this is an equation of the quadratic form where the coefficients are:

Since g, x^2, v₀^2 are positive (assuming x ≠ 0), the coefficient P(x) is always positive. This means that for a fixed x, the graph of y versus m is a parabola opening downwards. The maximum value of y occurs at the vertex of this parabola.

We can find the vertex by completing the square, and we get the following form:

As we discussed above in the property of envelope curve, the maximum value of y gives us the envelope Yₑ , so

The condition when y reaches maximum is when the square term in y(m) is zero, or:
We will come back to this condition of m in the next subsection using the derivative method.
This algebraic method of completing the square confirms the result we would get using calculus, showing that for any horizontal distance x, this Yₑ is the maximum possible height a projectile can achieve.

b. The Derivative Method (Formal Envelope Condition)

This method uses calculus. The general method for finding an envelope of a family of curves described by f(x,y,m)=0 (or, if y is an explicit function of x and m, y − F(x,m) = 0) is to solve the system of equations:

In our case, we have:
The partial derivative with respect to m is:

Setting this to zero to find the m that (for a fixed x) corresponds to a point on the envelope:

This is exactly the same condition as what we derived above to reach yₘₐₓ . Substituting this back into y = F(x,m) yields the same envelope:

c. The Discriminant Method: Where Solutions Merge

Rearrange the trajectory equation y = mx − \frac{g x^2}{2 v₀^2} (1 + m^2) into a quadratic equation in terms of m:

This is a quadratic equation of the form
with coefficient:

For a given point (x, y):

• If this quadratic equation has two distinct real solutions for m, it means there are two different launch angles θ (corresponding to m = \tanθ) that can reach the point (x, y). This point is inside the envelope.
• If it has no real solutions for m, the point (x, y) is unreachable. This point is outside the envelope.
• If it has exactly one real solution for m (a repeated root), the point (x, y) lies on the envelope. This is the boundary condition.

The condition for a quadratic equation to have exactly one real solution is that its discriminant Δ must be zero. So,

Assuming x ≠ 0 (the launch point x=0, y=0 is a special case on the envelope; if x=0, then y=0 is the only solution), we can divide by x^2:

Multiply by v₀^4:

Now, solve for y (which will be Y_E on the envelope):

Or:

Once again, we arrive at the same parabolic equation for the envelope of safety!

d. The Intersection of Neighboring Curves Method

Now one last method! As illustrated above, the envelope curve is the convergence of non-origin intersection of two infinitesimally close members of the family of projectile curves.

In this case, we consider two trajectories with slightly different parameters, m and m + δ m. Their equations are:

We find their intersection by solving this system of equations:

Simplifying this gives:

Now, let δ m → 0. The intersection point approaches the envelope, and the equation becomes:

This is the same condition to get the boundary as we saw over and over again. Substituting this value of m back into the original trajectory equation will, as we’ve seen, yield the envelope equation:

This method highlights the envelope as the locus of points where infinitesimally close members of the family of curves meet.

5. The Grand Unveiling: Summary and Reflections

It’s quite remarkable! Four different mathematical perspectives on the problem of finding the outer boundary of firework trajectories all converge on the exact same solution:

This is the equation of a parabola, often called Torricelli’s parabola or the parabola of safety.

a. The Beauty of Convergent Paths

More important than memorizing the formula is appreciating the underlying concepts. The fact that we can approach this problem by maximizing height (algebraically or with calculus), by looking for the boundary of solvability for launch angles, or by examining the meeting points of infinitesimally different trajectories—and all methods agree—is a testament to the consistency and elegance of mathematical physics. Each method provides a different shade of understanding to the nature of the envelope.

b. Significance of the Parabola of Safety

This isn’t just a neat mathematical result.

• Firework Safety: For pyrotechnicians, knowing this boundary (and adding appropriate safety factors for things like wind, dud shells, and projectile spread) is critical for determining safe viewing distances for the audience and protecting nearby structures.
• Military Applications: Historically, understanding projectile envelopes was crucial for artillery. The “parabola of safety” defines the maximum range and the boundary of the area that can be targeted by a gun with a given muzzle velocity. Any point outside this 3D paraboloid of revolution (if we rotate our 2D parabola around the vertical axis, assuming launches in all azimuthal directions) is safe from direct fire.
• Sports: Think of a baseball player trying to throw a ball as far as possible, or a shot putter. The optimal launch angle (around 45^∘ for maximum range on level ground, which is related to this envelope) comes from this same physics.

c. Viewing Fireworks with New Eyes (Again!)

In Part 1, we saw the instantaneous sphere of light. Now, as you watch those beautiful arcs, try to visualize not just the individual paths, but also that invisible, overarching parabolic cloak – the parabola of safety – that contains them all. It’s a silent testament to the predictable laws governing motion, even amidst the seemingly chaotic beauty of an explosion.

This concludes our deep dive into the physics of firework projectiles. Hopefully, you’ve gained a new appreciation for the intricate dance of math and physics that lights up our celebrations!

The Phillips Curve is still a useful, albeit imprecise, framework for understanding inflation, in particular its relationship with unemployment.

According to Wikipedia:

The Phillips curve is a single-equation econometric model, named after William Phillips, describing a historical inverse relationship between rates of unemployment and corresponding rates of rises in wages that result within an economy. Stated simply, decreased unemployment, (i.e., increased levels of employment) in an economy will correlate with higher rates of wage rises.

In my understanding, this is also correlated or augmented by inflation expectation. Inflation expectation is like a self-fulfilled prophecy, where public expectation of high inflation leads to more spending, and thus in turn, higher price inflation. In this case, inflation expectation can lead to steeper slope of the Philips curve, and vice versa.

What is noted here is that the Philips curve is becoming flat in the US, where current very low unemployment does not necessarily lead to a higher inflation. Former fed chair Ben Bernanke attributed this to Paul Volcker, who is widely credited with ending the high levels of inflation seen in the United States during the 1970s and early 1980s. As a result, people have high confidence that the Federal Reserve will not let the inflation go wild, as we have seen with the rate hike in 2018 due to risk with inflation due to rate hikes. This confidence leads to a lack of inflation expectation, and thus will not trigger the chain of events that drives up the inflation.

Quoted from Conan O’Brien’s final show at NBC.

Arnold Kling article on how socioeconomic elites maintains its status quo. More interesting is this comment below:

The issue is more then Asians despite having numbers, high IQ, and a powerful country of a billion plus that owns our government debt aren’t really “owners”. They are servants of the elite, dutifully making the hardest functions in our society work for UMC wages that barely pay the rent on the coasts, while others get rich and make the laws the govern society. At 40% they would be independent enough to challenge for leadership.

That’s why the 20% cap is everywhere, especially in the Northeast. It’s there at private elementary schools. It’s there in trying to shut down Asian magnet schools. It’s there in the bamboo ceiling. Enough to do the hard work, but not enough to lead!

Some say this is Asian personalities. That might be true in the sense that Asian traits aren’t what make you rich and powerful in society as structured today. However, we ought to ask if the people currently being made rich and powerful have traits that are good for the rest of us! Maybe a few more pragmatic, polite, conscientious, STEM and reality oriented Asians in elite leadership might be better for Americans. Getting power and using it for the countries best interests take different skill sets. Harvard only seems concerned with the former.

Quote from Making What Harvard Is About Transparent by Razib Khan.

The Left/liberal/progressive side engages in cant about “diversity”, when we all know they mean a very precise sort of diversity and a very particular type of background when they talk about “background.” But the Right/conservative side’s emphasis on merit and colorblindness strikes me as consciously blind to the fact that these institutions were always about shaping and grooming the elite and engaged in the game of reflecting and determining the American upper class. The Right/conservative project would abolish Harvard as we know it on a far deeper level than the Left/liberal/progressive posturing cultural radicalism, which at the end of the day has no problem bowing before neoliberal capital so long as lexical modifications are made.

If Asian Americans want to increase their proportion at Harvard, they have to follow the Jewish strategy and join the socio-political elite. If they don’t do that, then the Asian quota will persist in some way. The art of persuasion is really the exercise of power.

Six jobs that once paid more than the average wage and now pay less than the average:

1. jobs in the movie and music industries due to piracy, democratized digital tools,
2. Warehousing and storage.
3. Food production
4. Repair and maintenance
5. Auto sectors
6. Wood products

Above from Plant money indicator podcast.

The surge in October was because October was the first full month in which U.S. tariffs were in place on a full $250 billion of imports from China. I suppose we should see a similar graph from China side?

Sourced from WSJ today:

 

From George H. W. Bush’s inaugural address, and also quoted by George W. Bush at H.W. Bush’s funeral.

My friends, we are not the sum of our possessions. They are not the measure of our lives. In our hearts we know what matters. We cannot hope only to leave our children a bigger car, a bigger bank account. We must hope to give them a sense of what it means to be a loyal friend, a loving parent, a citizen who leaves his home, his neighborhood and town better than he found it.

What do we want the men and women who work with us to say? That we were more driven to succeed than anyone around us or that we stopped to ask if a sick child had gotten better and stayed a moment there to trade a word of friendship?