Problem 5.4.2.
We all know from experience that if you throw a ball straight up in the air, it will reach some maximum height. But suppose you throw the ball up twice as fast, would it go twice as high?
Section 5.4 Galileo Drops the Ball
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https://mathshistory.st-andrews.ac.uk/Biographies/Galileo/In the previous section we asserted — with no justification whatsoever — that the flight path of the Vomit Comet would be parabolic. Then, in Problem 5.3.1, we gave you the task of finding the constants \(B\text{,}\) and \(C\) for which the graph of the curve
\begin{equation*}
y=Ax^2+Bx+C
\end{equation*}
is the flight path of the Vomit Comet with an initial angle of elevation of \(45^\circ\text{.}\) You should have obtained
\begin{equation*}
y=Ax^2+x+7000.
\end{equation*}
Inside the airplane, we want what is called neutral buoyancy. This means that we don’t want any forces pushing us up or down, side to side, or back and forth relative to the airplane
The way to assure this is to force the airplane to fly along the same path it would take if it were unpowered and falling freely. At first it might seem like all we have to do is shut down the engine, but that won’t work because air resistance will prevent the plane from falling freely. The engines must be engaged to force the plane along the path it would follow naturally if there were no atmosphere and thus, no resistive forces. Also, it is incredibly dangerous to turn off the engines of an airplane while it is flying. In this section we’ll see why this path must be in the shape of a parabola.
Newton very famously said, “If I have been able to see further, it is by standing on the shoulders of giants,” and, in his \(1684\) paper on Calculus, Leibniz remarked that “Other very learned men have sought in many devious ways what someone versed in this calculus can accomplish in these lines as by magic.”
As we’ve seen Fermat, Descartes, and Roberval were three of the learned giants Newton and Leibniz were indebted to. Another was Professor Galilei of the University of Pisa, who is universally known and referred to by his first name, Galileo. As a professor of mathematics at Pisa, Galileo studied, among other things, how objects moved under the influence of gravity.
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https://mathshistory.st-andrews.ac.uk/Biographies/Galileo/We will be following Galileo’s lead to address such questions as:
- Suppose you throw a ball straight up into the air. How high will the ball go?
- How long will it take for it to hit the ground and what will its velocity be at impact?
- If you throw the ball up twice as fast. Will it go twice as high?
Since Galileo had neither the technology nor the mathematics to account for air resistance he ignored it. For the moment we will ignore it too.
The accepted theory of motion in Galileo’s time was Aristotle’s assertion that a heavier object would fall faster than a lighter object. Our common experience is that a hammer falls faster than a feather so this was an entirely reasonable thing to believe at the time. But we now know that this is because the resistance of the air slows the feather more than it does the hammer.
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https://en.wikipedia.org/wiki/AristotleIn \(1971\text{,}\) Apollo Astronaut David Scott confirmed this experimentally (see the video below) by dropping both a hammer and a feather on the surface of the moon. Because there is no air on the surface of the moon there is no air resistance, so the hammer and feather hit the surface of the moon at the same time.
Since he didn’t have access to the moon Galileo had to be clever instead. But Galileo was one of the group of new scientists who gathered experimental data and applied mathematical principles to theories. As a result of his experimental investigations Galileo proposed that in the absence of air all objects would fall at the same rate
How did he surmise that all objects would fall at the same rate?
Well actually, he didn’t. We need to be very careful in our use of language here. To say that “all objects fall at the same rate” is a bit sloppy. Since velocity is the rate of change of position it seems to say that all objects fall at the same velocity.
But we know that’s not true because the velocity of a falling object depends on how long it has been falling. At the moment you drop a ball its velocity is zero. Thereafter it gains velocity — it accelerates. Thus an object that has just been dropped is moving slowly while an object that has been falling for a while is moving at a faster rate.
What Galileo actually proposed was that the rate of change of the velocity — the acceleration — of all falling bodies is constant. But an object falling freely under the influence of Earth’s gravity gets moving pretty quickly and in Galileo’s day the tools available for measurement were very limited. So he slowed things down by letting small balls roll down a ramp as shown below.
Galileo assumed (correctly) that friction with the ramp would not significantly influence his measurements. When he did this he noticed that the balls always seemed to accelerate at a constant rate which depended only on the steepness of the ramp. More precisely, the acceleration of the rolling ball varied with the angle of descent. In modern terms we would say that the acceleration is a function of the angle of descent.
Referring to Figure 5.4.4, it is clear that when \(\theta=\pi/2\) the ball won’t move at all. So its acceleration is zero. When \(\theta=0\) the ball accelerates downward freely under the influence of gravity with no resistance Galileo measured the acceleration associated with steeper and steeper ramps (that is for \(\theta\) closer and closer to zero) obtaining a table of values similar to Table 5.4.5.
| \(\theta\) | Acceleration |
| \(\pi/3 \approx1.05\) | 4.9 |
| \(\pi/4 \approx0.79\) | 6.93 |
| \(\pi/6 \approx0.52\) | 8.49 |
| \(\pi/12 \approx0.26\) | 9.47 |
| \(\pi/24 \approx0.13\) | 9.72 |
| \(\pi/48 \approx0.07\) | 9.78 |
| \(\pi/100\approx0.03\) | 9.795 |
| \(\pi/200\approx0.02\) | 9.799 |
| \(\pi/300\approx0.01\) | 9.799 |
Notice that we are using radians to measure the angle. If you are unsure what a radian is you can refer to our brief review of Trigonometry in Section 6.1.
For scientific computations radian measure is usually simplest, so we will use it fairly consistently throughout this text. But degree measure is also common so it will occasionally make an appearance, like it did in Problem 5.3.1.
From the table it is clear that as \(\theta\) gets closer and closer to zero the acceleration is getting closer and closer to \(9.8.\) Thus Galileo deduced that when there is no ramp (when \(\theta =0\)) the velocity will increase each second by \(9.8\frac{\text{meters}}{\text{second}}.\)
That is, the velocity of an object falling under the influence of the earth’s gravity increases by \(9.8\) meters per second, each second. This is usually abbreviated as \(9.8\frac{\text{meters}}{\text{second}^2}\text{,}\) is commonly denoted by \(g\text{,}\) and is called the constant of acceleration due to gravity
Problem 5.4.6.
With a little Trigonometry you can deduce that \(g=9.8\frac{\text{meters}}{\text{second}^2}\) with only one measurement. Use the diagram and the first line of Table 5.4.5 to deduce that \(g=9.8.\)
Hint.
\(\cos\left(\pi/3\right)=1/2\text{.}\)
If we drop the object from some height, then its initial velocity is zero. After one second it will be falling at a rate of \(9.8\frac{\text{meters}}{\text{second}}\text{,}\) after two seconds, its velocity will be \(9.8\times2=19.6\frac{\text{meters}}{\text{second}}\text{,}\) etc. That is, as an object falls, its velocity increases at the constant rate of \(9.8 \frac{\text{meters}}{\text{second}}\) every second: \(v=9.8t.\)
The specific number that Galileo found, \(9.8\frac{\text{meters}}{\text{second}^2}\text{,}\) is an artifact of the units we use to measure distance (meters) and time (seconds) and the fact that we are on the surface of a particular planet. If we measure distance in feet instead then at the surface of the earth a falling object will accelerate at \(32 \frac{\text{feet}}{\text{second}^2}\text{.}\) If we go to the surface of the moon, it will accelerate at \(1.6
\frac{\text{meters}}{\text{second}^2}\text{.}\)
Problem 5.4.7.
Show that if we measure distance in feet, the acceleration constant on the moon is approximately \(5.2\frac{\text{feet}}{\text{second}^2}.\)
The general situation is this: If a falling object’s velocity, in meters per second \(\left(\frac{\text{meters}}{\text{second}}\right)\text{,}\) is changing at a constant rate of \(r\) meters per second per second \(\left(\frac{\text{meters}}{\text{second}^2}\right)\text{,}\) and \(t\) is the number of elapsed seconds then \(v=rt\text{.}\) From Galileo’s work we know that at the surface of the Earth \(r=g=9.8\) so
\begin{equation}
v=9.8t\tag{5.4}
\end{equation}
We can check that this is reasonable by looking at the units of measurement. Since the acceleration \(r,\) is measured in \(\frac{\text{meters}}{\text{seconds}^2}\) and time is measured in seconds, when they are multiplied a symbolic cancellation gives
\begin{equation*}
\frac{\text{meters}}{\text{second}^2}\cdot \text{second} = \frac{\text{meters}}{\text{second}\cdot \cancel{\text{second}}}\cdot \cancel{\text{second}} = \frac{\text{meters}}{\text{second}}
\end{equation*}
or meters per second; the units used to measure velocity. Notice that equation (5.4) is independent of the weight (or mass) of the falling object. Thus Galileo hypothesized that under the influence of gravity alone all objects accelerate downward at the same rate. Galileo’s hypothesis — which contradicts Aristotle’s claim — came from both his experimental evidence and a famous thought experiment he described in his book On Motion (1590).
The experiment runs as follows: Imagine that two objects, one light and one heavy, are connected to each other by a string and we drop them from a great height, say the top of the Tower of Pisa. If we assume heavier objects do indeed fall faster than lighter ones the string will soon pull taut as the lighter object retards the fall of the heavier object.
So, the system taken as a whole will fall more slowly than the heavier object alone.
But the system as a whole is heavier than either individual object. So according to Aristotle, the system taken as a whole should fall faster than the heavier object.
This contradiction leads inexorably to the conclusion that our initial assumption — Aristotle’s assertion that heavier objects accelerate faster — must be false.
According to legend Galileo tested his hypothesis by dropping balls of different weights from the top of the Tower of Pisa, but that is almost certainly pure legend. He never actually did this. Probably.
Galileo set out his ideas about falling objects in his last book Discorsi e Dimostrazioni Matematiche intorno à due nuoue Scienze (Discourses and Mathematical Demonstrations Concerning Two New Sciences) (1638). This was the last of Galileo’s many scientific works
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https://old.maa.org/press/periodicals/convergence/mathematical-treasure-galileos-two-new-sciencesThe two sciences referred to in the title were the science of motion, which became the foundation of modern physics, and the science of materials and construction, an important contribution to engineering.
Suppose an object falls \(56\) meters from the top of the Tower of Pisa to the ground. Can we determine how fast it is moving when it strikes the ground?
Recall that the formula \(v=gt\) tells us the ball’s velocity at any time \(t\text{.}\) So if \(t_g\) is the amount of time it takes for the ball to reach the ground then the velocity at the end would be \(v=gt_g\text{.}\) But how do we find out exactly when the ball strikes the ground?
To see the difficulty, suppose we dropped the ball from a height of \(9.8\) meters. Would it take \(1\) second for the ball to hit the ground?
Clearly not.
To reach the ground after \(1\) second the ball would have to average \(9.8\frac{\text{meters}}{\text{second}}\) during the entirety of that first second. But at first it is not moving at all \((\text{velocity} =0)\text{.}\) After one second its velocity has increased to \(9.8 \frac{\text{meters}}{\text{second}}\text{.}\) So for the entire duration of that first second the ball’s velocity is less than \(9.8
\frac{\text{meters}}{\text{second}}\text{.}\) Thus after \(1\) second it has not yet hit the ground because it was never going fast enough to do so. But exactly how far it has fallen still isn’t clear.
Reasoning similarly, at the beginning of the next second the ball is already falling at \(9.8\frac{\text{meters}}{\text{second}}\) and thereafter it’s velocity increases, so it falls at least \(9.8\) meters during the next second, in addition to the distance it fell during the first. So, while we cannot yet tell exactly how far it falls during either second, we can say that,
- it will not hit the ground during the first second, and
- it definitely will hit the ground sometime during the next second.
If we had an expression for the ball’s position, \(p\text{,}\) similar to equation (5.4), we would be in much better shape. Galileo was able to do this the tools of Calculus. Since learning to use those tools is our purpose we will take the easier route of Calculus.
