WHEN AN APPLE FALLS

Would it make sense to you if you saw an apple fall from the tree but only halfway? What type of a universe would that be if falling apples randomly snapped out of existence and reappeared somewhere else? Luckily our universe is not like that. The only incomprehensible thing about our universe, following Einstein, is that it is comprehensible. We can attempt to discover Her hidden laws, hoping that one day, we will have our hands on some singular equation that could explain everything. But before we hop on to the big questions, let us start somewhere simple. Let us start with Newton's apple. 

To say with a personal touch, I find it mildly amusing to picture one of humanity's greatest mathematical geniuses, Sir Isaac Newton, resting under an apple tree when an apple falls on his head, and he runs wild through the open fields of Lincolnshire shouting gravity! There may be very little truth to the above story of the apple striking his head, but as he said, his intellect got stirred in the wake of falling apples. He questioned what mysterious entity made an apple always fall perpendicular to the tree instead of falling sideways or going up. If the Earth had a tendency to attract objects towards its centre through gravity, then Newton asked, what kept the moon from crashing onto Earth? What kept the Sun and all the heavenly stars from falling? 

123 many apples fell
And gravity it is we can tell.
Image Credits: Public Domain

Newton saw gravity not as an attractive force inherent to Earth but as a universal force inherent to all matter. He argued that it is not only the Earth that attracts the apple, but the apple also attracts the Earth, wherein the power of attraction lies in their constituent masses and how far they are from each other. A heavier object would attract more strongly than a lighter one. Thus in 1687, in his Philosophiæ Naturalis Principia Mathematica, Newton published the final form of the law of universal gravitation. The law states that all matter possesses the inherent property of attracting every other matter with a force that is proportional to the product of their masses and inversely proportional to the square of the straight line distance separating the two. In a better language, every point mass or a point particle attracts every other point mass by a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance separating the pair, with the force acting along the line intersecting the two point masses or along the line joining their centre of masses for larger objects. This whole statement can be expressed as  

, where F is the force between the two point masses m₁ and m₂, r² is the distance separating the two point masses, and G is the universal gravitational constant. 

Following Kepler's laws, Newton argued that some force must keep the planets and their moons in perfect orbits; otherwise, they would fly off. Using the laws of motion and universal gravitation, Newton solved the central problem of celestial motion - what made the planets move the way they do? Looking at the moon, he realised that our celestial spectacle is continually being attracted towards the Earth and is essentially falling, but at the same time, possessing a tangential velocity component, the moon is able to revolve around the Earth in a fixed orbit. Newton saw that the force which made an apple fall was the same force responsible for celestial motion. The moon does not fall on us because it is falling - in a orbit; it is falling in a orbit!

In this article, I am not going to talk at length about celestial dynamics. I will put that in another article. For now, my sole focus will be on the free fall of objects under a constant gravitational field, i.e., the simplest case of falling apples. However, the reader may ask why I began this article stating the law of universal gravity. I must ask you to be patient, for I will take you to a beautiful discovery relating to free fall. 

Let there be a body of mass m at some height h above the surface of the Earth. We must bear in mind that this height h is much much less than the radius of the Earth so as to neglect the variation of g with increasing height from the surface. A non-rotating, perfectly spherical body of uniform mass density will produce a uniform gravitational field of equal magnitude at all points on its surface and hence the same gravitational acceleration at all points on the surface. But since Earth is rotating, it deviates appreciably from happening to be a perfect sphere, being flatter at the poles and bulging at the equator. As a result, the acceleration due to gravity at all points on the surface does not remain constant. At the surface, the acceleration due to gravity (symbolized by g) is approximately taken to be 9.80665 m/s² or simply 9.8 m/s² for convenience. To make our calculations simpler, we can neglect Earth's rotation and curvature.. More importantly, we are considering free fall motion in a perfect vacuum without any air resistance because when in air, the falling body will experience a retarding (opposing) force, and its free fall nature will change drastically. With a suitable choice of coordinates as shown in the following figure, and taking the downward direction to be positive,

we can write the equation of motion for perfect (natural) free fall of a body falling under a constant gravitational field and under no air resistance as follows:

Equation 1.1

The boundary conditions are such that at t = 0, the body starts falling with 0 initial velocity, i.e., it essentially falls from a state of rest. Let us assume that the body falls for a height of t seconds before hitting the ground, and as such, at t = 0, z(t) = 0, where the latter denotes the displacement of the body in t seconds. 

Performing the first integration and applying the necessary boundary conditions, we arrive at

Equation 1.2a

, which gives us the velocity of the falling body at t seconds. The second successive integration yields the distance the particle falls through in t seconds, i.e., we have,  

Equation 1.2b

If the particle starts off with an initial velocity of vᵢ, then the above pair of equations take the following form.

Equation 1.3a

Equation 1.3b

These are our basic free fall equations, from which we can easily find out the related quantities such as the time of flight if the height is known and vice versa, displacement at the nth second and so on. 

Now that we have arrived at the necessary kinematics of free fall, we can take a breather. When we first learn about free fall in high school or first-year college physics, we may fail to fully comprehend the depth of Newtonian mechanics. Let us take a closer look at equation 1.2a. It shows that the velocity of a freely falling body is nothing but a linear function of time multiplied by the gravitational acceleration. When a body is allowed to fall from rest, at 1 second, its velocity will be g, then 2g at 2 seconds, 3g at 3 seconds, and so on. It turns out that during perfect free fall, once again, under no air resistance and constant gravitational acceleration, the velocity of the falling body progressively increases with each passing second. If a body is dropped from such a height that it hits the ground in 10 seconds, its velocity will be 98 m/s or 352.8 km/h. From Newton's second law of motion, we find that in free space (vacuum), if a constant force is applied to a particle of mass m, it will continue accelerating for as long as the force is being applied. As such, if we keep on applying a force, then the particle will accelerate indefinitely and to infinity. In a like manner, if it were possible for a particle to fall straight down indefinitely in a constant gravity field, the particle would accelerate without bounds. Theoretically, it will accelerate to infinity because Newton's laws imposes no restriction for such an event. However, in reality, the gravitational field of any object, even though it extends to infinity, its influence, i.e., the attractive power, falls inversely to the square of the distance. Since g varies with increasing distance from the Earth's surface, at greater distances, it becomes almost negligible as per the following expression. 

Had g remained constant for h → ∞, distant stars would come tumbling down on Earth with an infinite velocity and considering all the infinities, the universe would not have existed at all. 

Let us take a second look at equation 1.2a. It turns out that when in perfect free fall, the velocity of the falling becomes independent of mass. No matter how heavier the body is, whatever its shape, size, density, or surface characteristics, whether it is a peacock feather or a fully healthy T. Rex. brought back to life from the fossils, while in perfect free all, they fall with the same velocity independent of their constituent masses. Velocity and displacement possess only a linear and quadratic dependence on time only. One of the best places where we can test the validity of our above relation, is the moon. The moon has so tenuous an atmosphere that we can take it as a nearly perfect vacuum environment where we can drop a feather and a hammer. Just like our equation 1.2a predicts, the feather and the hammer will fall with the same velocity drawn towards the centre of the moon by gₘₒₒₙ, where the latter denotes the gravitational acceleration of the moon. Luckily we do not need to perform this experiment right away, for astronomer David Scott during the Apollo 15 mission, performed this experiment as seen in the following video. 

 

This perfectly linear time dependence of velocity drastically changes when we plug in the factor of air resistance. Air is matter, and by that virtue, it offers resistance to free fall. What happens is that when falling through the air (i.e., the atmosphere), a body experiences two forces, viz., the downward acceleration due to gravity that pulls the body towards the surface and an oppositely directed resistance force known as the viscous drag of air. To fully understand the concept of viscous drag, we need to bring to the table the principles of fluid mechanics, and for that, I intend to write a separate article. But for now, the viscous drag arises because air tries to inhibit the motion of a body through itself. The viscous drag of air depends upon the mass of the falling object, its density, surface area and associated surface features, the height from which it is falling, and whether it possesses any initial velocity. Due to air resistance, the velocity of the falling object, as we will see in our next article, follows an exponentially decaying relation to time. Therefore, it assumes a terminal velocity, which is the maximum velocity of free fall under air resistance. When we drop a feather and a coin, we have all noticed that the coin falls faster and hits the ground much before the feather. Interestingly, if we put the coin and feather in a vacuum chamber, they will fall together. It is not that while in the air, the feather experiences a lesser pull of gravity. Both the coin and the feather are pulled downwards with the same g. It is air resistance that slows down the feather. More precisely, the feather is lighter than the coin, and because it covers a greater surface area, it quickly reaches its terminal (limiting velocity). Upon reaching the terminal velocity, the body does not accelerate further because the upward force of air balances the downward pull of gravity and depending on the nature of the falling body, it will with a lower velocity than in the case of perfect free fall. 

Coming to this length, it is time we paid our homage to Galileo Galilei for whatever kinematical equations we have used so far, and our arguments have been the singular work of the famed Italian mathematician. Since the time of Aristotle, people unquestionably believed that heavier objects fall faster than lighter ones. As per Aristotle's teachings, bodies fall proportional to their weights, viz., a ten times massive body ought to fall ten times faster. Although at first impression, Aristotle may sound right because, in the case of a feather and a coin, we have all noticed that the latter being heavier falls faster. However, Galileo showed that this is not a strong argument. He investigated heavily into the nature of freely falling bodies, either vertically or across various inclined planes, projectiles and over curved trajectories, from which he concluded that all bodies irrespective of their mass fall with the same velocity and as such, they accelerate equally under Earth's gravity (or gravity in general). Legend has it that he dropped two lead balls from the Leaning Tower of Pisa, one ten times heavier than the other and to the much dismay of the nearby spectators, showed them that the balls fall equally. More on this head can be read in his book, On Two New Sciences. With the help of inclined planes and water clocks, he was also the first to provide a measure of the acceleration due to gravity of the Earth. All this was in stark contrast to Aristotle's teachings. Plus, being aware of air resistance, Galileo introduced the notion of perfect, natural free fall in the absence of air, i.e., in a vacuum. If only he could create a vacuum chamber, he would have witnessed a wonderous discovery. 

                                                                                                                               To be continued...