## Why does a ball fall only about 5 meters in its first second of falling?

### After one second of falling, a ball’s velocity is about 10 meters per second downward. So why does it travel only about 5 meters downward during that one second?

If the ball traveled steadily at 10 meters per second downward for one second, it would travel 10 meters downward. But a falling ball does not move steadily. Instead, it accelerates downward and its velocity changes with time. When dropped from rest, its velocity starts at 0 and steadily increases to about 10 meters per second downward after 1 second of falling. Its average velocity during that 1 second interval is only about 5 meters per second (the average of 0 and 10 meters per second) downward. Therefore, the ball only moves downward about 5 meters.

## Why does a sheet of paper fall faster when it’s above a falling book?

### Why does a sheet of paper on top of a large book fall and land at the same time as the book?

By itself, a falling sheet of paper experiences severe air resistance as it moves downward through stationary air. It soon reaches a small terminal velocity — the downward speed at which the upward force of air resistance cancels its downward weight and it stops accelerating downward. The falling book protects the sheet from that air resistance, so the sheet can fall unimpeded.

All objects that move downward through stationary air experience upward air resistance forces, but heavy compact objects (e.g., books) are less affected by those forces than light, fluffy ones (e.g., sheets of paper). Although a book falls mostly unimpeded through stationary air, it does affect the air it encounters. Most importantly, it drags a pocket of air with it. The air just above the book is moving downward at approximately the book’s velocity. When a sheet of paper is located in this special region of downward-moving air, it can fall with the book and experience virtually no forces due to the air.

This behavior is known as drafting and is important in many types of races, including running, bicycling, skating, and swimming. For example, a bicyclist drags a pocket of air with her and a second bicyclist following close behind her and located in that pocket of forward-moving air experiences less air resistance. Consequently, a group of bicyclists traveling in a tight line makes easier forward progress through stationary air than separated bicyclists each fighting air resistance in their own. The lead bicyclist in the line gets the air moving for the others and exhausts first, so they typically take turns as the lead bicyclist. Some races forbid drafting because it provides such an advantage for the following bicyclist.

Drafting while skydiving is actually dangerous and is normally avoided. A descending skydiver creates a pocket of downward-moving air above them and when a second skydiver enters that pocket of downward-moving air, the upward air resistance decreases dramatically. The second skydiver suddenly accelerates downward due to weight and can drop onto the first skydiver. This is usually a bad idea and can lead to disaster. Similarly, the sheet of paper drops onto the book below it, but the only possibility of injury is if they land on your foot.

## Which ball lands first: one thrown up or one thrown forward?

### If a ball is thrown horizontally, will it still land at the same moment as a ball thrown straight up?

The ball thrown horizontally will land first. Its velocity immediately begins to develop a downward component and it moves downward faster and faster until lands on the ground. In contrast, the ball throw straight up starts with an upward component to its velocity and it rises to a peak height before it begins to develop a downward component to its velocity. Its travel time to the ground is much longer.

A more sophisticated way to look at this question is to recognize that you can separate the horizontal and vertical motions of a falling ball. Since the ball’s weight is purely vertical, it has no effect on the ball’s horizontal motion. The ball’s horizontal motion is that of a coasting: the ball’s horizontal component of velocity never changes. If the ball had any forward component of velocity when you released it, that ball will make steady progress in its forward direction.

The ball’s vertical motion, however, is that of falling: the ball’s vertical component of velocity changes with time as the ball accelerates downward at the acceleration due to gravity. The ball thrown horizontally begins its fall with zero vertical component to its velocity. It’s effectively falling from rest in terms of its vertical motion. It drops faster and faster and soon lands on the ground. The ball thrown upward, however, beings its fall with an upward vertical component to its velocity. It travels upward at first, rising more and more slowly until it reaches its peak height momentarily, and then it descends more and more quickly until it reaches the ground. That second motion, up and then down, takes far longer than the first motion, down only.

## How can you calculate the position of a falling ball?

### When you drop a ball, its position changes in a complicated way. How would you calculate that position?

When you drop a ball, its altitude decreases by larger and larger increments as the seconds pass. If we call the altitude from which you drop it zero, then its altitude after 1 second is -4.9 m (-4.9 meters or about -16 feet), after 2 seconds is -19.6 m, and after 3 seconds is – 44.1 m. Here is one way to calculate those values.

First, note that the ball is accelerating downward steadily at 9.8 m/s2. The ball’s initial velocity was zero, so its velocity after falling for time t is 9.8 m/s2 * t downward.

Next, let’s find the ball’s average velocity while falling for time t. The ball’s velocity has been changing steadily from 0 when you dropped it to 9.8 m/s2 * t downward after falling for time t, so it’s average velocity is simply the average of those two individual values: 0 and 9.8 m/s2 * t downward. That average is 4.9 m/s2 downward.

Lastly, let’s determine how far downward the ball has traveled after falling for time t. Since it’s average velocity was 4.9 m/s2 * t downward and it has traveled for time t with that average velocity, its change in position is 4.9 m/s2 * t downward * t or simply 4.9 m/s2 * t2 downward. As you can see, its change in position is proportional to the square of its fall time t. With each passing second, it is moving downward faster and covering more distance. As stated above, its altitude after 1 second of falling is -4.9 m, after 2 seconds of falling is -19.6 m, and after 3 seconds of falling is -44.1 m.

## How dangerous is a penny falling from the Empire State Building?

#### If a penny fell from the Empire State Building, could it actually punch a hole in the sidewalk?

A famous urban legend states that a penny dropped from the top of the Empire State Building will punch a hole in the sidewalk below. Given the height of the building and the hardness of the penny, that seems like a reasonable possibility. Whether it’s true or not is a matter that can be determined scientifically. Before we do that, though, let’s get some background.

Falling rocks can be dangerous and, the farther they fall, the more dangerous they become. Falling raindrops, snowflakes, and leaves, however, are harmless no matter how far they fall. The distinction between those two possibilities has nothing to do with gravity, which causes all falling objects to accelerate downward at the same rate. The difference is entirely due to air resistance.

Air resistance—technically known as drag—is the downwind force an object experiences as air moves passed it. Whenever an object moves through the air, the two invariably push on one another and they exchange momentum. The object acts to drag the air along with it and the air acts to drag the object along with it, action and reaction. Those two aerodynamic forces affect the motions of the object and air, and are what distinguish falling snowflakes from falling rocks.

Two types of drag force affect falling objects: viscous drag and pressure drag. Viscous drag is the friction-like effect of having the air rub across the surface of the object. Though important to smoke and dust particles in the air, viscous drag is too weak to affect larger objects significantly.

In contrast, pressure drag is strongly affects most large objects moving through the air. It occurs when airflow traveling around the object breaks away from the object’s surface before reaching the back of the object. That separated airflow leaves a turbulent wake behind the object—a pocket of air that the object is effectively dragging along with it. The wider this turbulent wake, the more air the object is dragging and the more severe the pressure drag force.

The airflow separation occurs as the airflow is attempting to travel from the sides of the object to the back of the object. At the sides, the pressure in the airflow is especially low due as it bends to arc around the sides. Bernoulli’s equation is frequently invoked to help explain the low air pressure near the sides of the object. As this low-pressure air continues toward the back of the object, where the pressure is much greater, the airflow is moving into rising pressure and is pushed backward. It is decelerating.

Because of inertia, the airflow could be expected to reach the back of the object anyway. However, the air nearest the object’s surface—boundary layer air—rubs on that surface and slows down. This boundary layer doesn’t quite make it to the back of the object. Instead, it stops moving and consequently forms a wedge that shaves much of the airflow off of the back of the object. A turbulent wake forms and the object begins to drag that wake along with it. The airflow and object are then pushing on one another with the forces of pressure drag.

Those pressure drag forces depend on the amount of air in the wake and the speed at which the object is dragging the wake through the passing air. In general, the drag force on the object is proportional to the cross sectional area of its wake and the square of its speed through the air. The broader its wake and the faster it moves, the bigger the drag force it experiences.

We’re ready to drop the penny. When we first release it at the top of the Empire State Building, it begins to accelerate downward at 9.8 meters-per-second2—the acceleration due to gravity—and starts to move downward. If no other force appeared, the penny would move according to the equations of motion for constant downward acceleration, taught in most introductory physics classes. It would continue to accelerate downward at 9.8 meters-per-second2, meaning that its downward velocity would increase steadily until the moment it hit sidewalk. At that point, it would be traveling downward at approximately 209 mph (336 km/h) and it would do some damage to the sidewalk.

That analysis, however, ignores pressure drag. Once the penny is moving downward through the air, it experiences an upward pressure drag force that affects its motion. Instead of accelerating downward in response to its weight alone, the penny now accelerates in response to the sum of two force: its downward weight and the upward drag force. The faster the penny descends through the air, the stronger the drag force becomes and the more that upward force cancels the penny’s downward weight. At a certain downward velocity, the upward drag force on the penny exactly cancels the penny’s weight and the penny no longer accelerates. Instead, it descends steadily at a constant velocity, its terminal velocity, no matter how much farther drops.

The penny’s terminal velocity depends primarily on two things: its weight and the cross sectional area of its wake. A heavy object that leaves a narrow wake will have a large terminal velocity, while a light object that leaves a broad wake will have a small terminal velocity. Big rocks are in the first category; raindrops, snowflakes, and leaves are in the second. Where does a penny belong?

It turns out that a penny is more like a leaf than a rock. The penny tumbles as it falls and produces a broad turbulent wake. For its weight, it drags an awful lot of air behind it. As a result, it reaches terminal velocity at only about 25 mph (40 km/h). To prove that, I studied pennies fluttering about in a small vertical wind tunnel.

Whether the penny descends through stationary air or the penny hovers in rising air, the physics is the same. Of course, it’s much more convenient in the laboratory to observe the hovering penny interacting with rising air. Using a fan and plastic pipe, I created a rising stream of air and inserted a penny into that airflow.

At low air speeds, the penny experiences too little upward drag force to cancel its weight. The penny therefore accelerated downward and dropped to the bottom of the wind tunnel. At high air speeds, the penny experienced such a strong upward drag force that it blew out of the wind tunnel. When the air speed was just right, the penny hovered in the wind tunnel. The air speed was then approximately 25 mph (40 km/h). That is the terminal velocity of a penny.

The penny tumbles in the rising air. It is aerodynamically unstable, meaning that it cannot maintain a fixed orientation in the passing airstream. Because the aerodynamic forces act mostly on the upstream side of the penny, they tend to twist that side of the penny downstream. Whichever side of the penny is upstream at one moment soon becomes the downstream side, and the penny tumbles. As a result of this tumbling, the penny disturbs a wide swath of air and leaves a broad turbulent wake. It experiences severe pressure drag and has a low terminal velocity.

The penny is an example of an aerodynamically blunt object—one in which the low-pressure air arcing around its sides runs into the rapidly increasing pressure behind it and separates catastrophically to form a vast wake. The opposite possibility is an aerodynamically streamlined object—one in which the increasing pressure beyond the object’s sides is so gradual that the airflow never separates and no turbulent wake forms. A penny isn’t streamlined, but a ballpoint pen could be.

Almost any ballpoint pen is less blunt than a penny and some pens are approximately streamlined. Moreover, pens weigh more than pennies and that fact alone favors a higher terminal velocity. With a larger downward force (weight) and a smaller upward force (drag), the pen accelerates to a much greater terminal velocity than the penny. If it is so streamlined that it leaves virtually no wake, like the aerofoil shapes typical of airplane components, it will have an extraordinarily large terminal velocity—perhaps several hundred miles per hour.

Some pens tumble, however, and that spoils their ability to slice through the air. To avoid tumbling, a pen must “weathervane”—it must experience most of its aerodynamic forces on its downstream side, behind its center of mass. Arrows and small rockets have fletching or fins to ensure that they travel point first through the air. A ballpoint pen can achieve that same point-first flight if its shape and center of mass are properly arranged.

Almost any ballpoint pen dropped into my wind tunnel plummeted to the bottom. I was unable to make the air rise fast enough to observe hovering behavior in those pens. Whether they would tend to tumble in the open air was difficult to determine because of the tunnel’s narrowness. Nonetheless, it’s clear that a heavy, streamlined, and properly weighted pen dropped from the Empire State Building would still be accelerating downward when it reached the sidewalk. Its speed would be close to 209 mph at that point and it would indeed damage the sidewalk.

As a final test of the penny’s low terminal velocity, I built a radio-controlled penny dropper and floated it several hundred feet in the air with a helium-filled weather balloon. On command, the dropper released penny after penny and I tried to catch them as they fluttered to the ground. Alas, I never managed to catch one properly in my hands. It was a somewhat windy day and the ground at the local park was uneven, but that’s hardly an excuse—I’m simply not good at catching things in my hands. Several of the pennies did bounce off my hands and one even bounced off my head. It was fun and I was more in danger of twisting my ankle than of getting pierced by a penny. The pennies descended so slowly that they didn’t hurt at all. Tourist below the Empire State Building have nothing fear from falling pennies. Watch out, however, for some of the more streamlined objects that might make that descent.

## What becomes of a log’s gravitational energy when you burn it?

#### If one takes firewood to the top of a hill and burns it there, does the firewood’s gravitational energy disappear? — V

When you carry the firewood up the hill, you transfer energy to it and increase its gravitational potential energy. When you then burn the wood, you seem to make this energy disappear. After all, there doesn’t appear to be any difference between burning wood in the valley and burning wood on the top of the hill. The wood is gone either way.

But appearances can be deceiving. Since energy is a conserved quantity, the energy that you invest in the firewood can’t disappear. It simply becomes difficult to find because it is dispersed in the burned gases that were once the wood.

To find that energy, imagine compressing the burned gases into a small container to make their weight more noticeable and reduces buoyant effects due to the atmosphere. You could then carry those burned gases, which include all of the firewood’s atoms, back down the hill. As you descended, the container of burned gases would transfer its gravitational potential energy to you.

I’ve swept a number of details under the rug, such as the fact that many of the oxygen atoms in your container were originally part of the atmosphere rather than the log. But even when all those details are taken into account, the answer is the same: the firewood’s gravitational energy doesn’t disappear, it just gets more difficult to find.

## How come planets are spherical, albeit with somewhat flattened poles?

#### How come planets are spherical, albeit with somewhat flattened poles? — DB

The answer is gravity. Gravity smashes the planets into spheres. To understand this, imagine trying to build a huge mountain on the earth’s surface. As you begin to heap up the material for your mountain, the weight of the material at the top begins to crush the material at the bottom. Eventually the weight and pressure become so great that the material at the bottom squeezes out and you can’t build any taller. Every time you put new stuff on top, the stuff below simply sinks downward and spreads out. You can’t build bumps bigger than a few dozen miles high on earth because there aren’t any materials that can tolerate the pressure. In fact, the earth’s liquid core won’t support mountains much higher than the Himalayas—taller mountains would just sink into the liquid. So even if a planet starts out non-spherical, the weight of its bumps will smash them downward until the planet is essentially spherical.

The flattened poles are the result of rotation—as the planet spins, the need for centripetal (centrally directed) acceleration at its equator causes its equatorial surface to shift outward slightly, away from the planet’s axis of rotation. The planet is therefore wider at its equator than it is at its poles.

## How do you convert a measurement in liters per second into one in gallons per mi…

#### How do you convert a measurement in liters per second into one in gallons per minute? — MG

Converting units is always a matter of multiplying by 1. But you must use very fancy versions of 1, such as 60 seconds/1 minute and 1 gallon/3.7854 liters. Since 60 seconds and 1 minute are the same amount of time, 60 seconds/1 minute is 1. Similarly, since 1 gallon (U.S. liquid) and 3.7854 liters are the same amount of volume, 1 gallon/3.7854 liters is 1. So suppose that you have measured the flow of water through a pipe as 283 liters/second. You can convert to gallons/minute by multiplying 283 liters/second by 1 twice: (283 liters/second)(60 seconds/1 minute)(1 gallon/3.7854 liters). When you complete this multiplication, the liter units cancel, the second units cancel, and you’re left with 4,486 gallons/minute.

## If light has no mass, then how can it be affected by gravity? What property of l…

#### If light has no mass, then how can it be affected by gravity? What property of light is gravitational force acting on? — DM

At low speeds, mass and energy appear to be separate quantities. Mass is the measure of inertia and can be determined by shaking an object. Energy is the measure of how much work an object can do and can be determined by letting it do that work. Conveniently enough, the object’s weight—the force gravity exerts on it—is exactly proportional to its mass, which is why people carelessly interchange the words “mass” and “weight,” even though they mean different things.

But when something is moving at speeds approaching the speed of light, mass and kinetic energy no longer separate so easily. In fact, the relativistic equations of motion are more complicated than those describing slow objects and the way in which gravity affects fast objects is more complicated than simply giving them “weight.”

Overall, you can view the bending of light by gravity in one of two ways. First, you can view it approximately as gravity affecting not on mass, but also energy so that light falls because its energy gives it something equivalent to a “weight.” Second, you can view it more accurately as the bending of light as caused by a change in the shape of space and time around a gravitating object. Space is curved, so that light doesn’t travel straight as it moves past gravitating objects—it follows the curves of space itself. The second or Einsteinian view, which correctly predicts twice as much bending of light as the first or Newtonian view, is a little disconcerting. That’s why it took some time for the theory of general relativity to be widely accepted. (Thanks to DP for pointing out the factor of two.)

## I have been trying to get information on what causes strange gravity areas to ex…

#### I have been trying to get information on what causes strange gravity areas to exist…Walking on walls, water rolling uphill, etc. There are a number of such places advertised in the United States and elsewhere but are they optical illusions or for real? — MW

These purported gravitational anomalies are just illusions. Because gravity is a relatively weak force, enormous concentrations of mass are required to create significant gravitational fields. Since it takes the entire earth to give you your normal weight, the mass concentration needed to cancel or oppose the earth’s gravitation field in only one location would have to be extraordinary. While objects capable of causing such bizarre effects do exist elsewhere in our universe (e.g. black holes and neutron stars), there fortunately aren’t any around here. As a result, the strength of the gravitational field at the earth’s surface varies less than 1% over the earth’s surface and always points almost exactly toward the center of the earth. Any tourist attraction that claims to have gravity pointing in some other direction with some other strength is claiming the impossible.