Problem 60 - Calculus

Question

When Galileo was formulating the laws of motion, he considered the motion of a body starting from rest and falling under gravity. He originally thought that the velocity of such a falling body was proportional to the distance it had fallen. What do the experimental data in Table 1.6 tell you about Galileo's hypothesis? What alternative hypothesis is suggested by the two sets of data in Table 1.6 and Table \(1.7 ?\) $$\begin{array}{c|c|c|c|c|c}\hline \text { Distance (ft) } & 0 & 1 & 2 & 3 & 4 \\\\\hline \text { Velocity (ft/sec) } & 0 & 8 & 11.3 & 13.9 & 16 \\\\\hline\end{array}$$ $$\begin{array}{c|c|c|c|c|c}\hline \text { Time (sec) } & 0 & 1 & 2 & 3 & 4 \\\\\hline \text { Velocity (ft)sec) } & 0 & 32 & 64 & 96 & 128 \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

The data suggest velocity is proportional to time, not distance.

Step 1: Analyze Galileo's Hypothesis

Galileo originally hypothesized that a falling body's velocity is proportional to the distance it has fallen. This means that velocity should increase linearly with distance. Therefore, we should expect a consistent incremental increase in velocity as distance increases.

Step 2: Examine Table 1.6

Table 1.6 provides data on velocity concerning distance. Calculate the changes in velocity: - From 0 to 1 ft: 8 ft/sec increase, - From 1 to 2 ft: 3.3 ft/sec increase, - From 2 to 3 ft: 2.6 ft/sec increase, - From 3 to 4 ft: 2.1 ft/sec increase. The increase in velocity is not consistent or proportional to the distance fallen.

Step 3: Propose an Alternative Hypothesis

Based on Table 1.6, the hypothesis of proportionality with distance is incorrect. Instead, the velocity increases at a decreasing rate, suggesting a different relationship.

Step 4: Examine Table 1.7

Table 1.7 provides time-dependent velocity data, showing linear increments: - From 0 to 1 sec: 32 ft/sec increase, - From 1 to 2 sec: 32 ft/sec increase, - From 2 to 3 sec: 32 ft/sec increase, - From 3 to 4 sec: 32 ft/sec increase. The velocity increases consistently with time, suggesting a linear relationship.

Step 5: Formulate the New Hypothesis

Based on Table 1.7, an alternative hypothesis is that the velocity of a falling body is directly proportional to the time elapsed, not the distance fallen. This aligns with the concepts of uniform acceleration under gravity.

Key Concepts

Velocity

Velocity is an essential aspect of understanding how objects move. When discussing Galileo's laws of motion, velocity describes how fast an object falls as it moves from rest. Imagine the world in Galilean times; it was assumed that velocity increased in direct proportion to the distance an object fell. This would mean every additional foot an object fell came with a predictable increase in speed. However, the experimental data show that's not quite how things work.
In reality, velocity is not just a simple function of distance. In our exercise, Table 1.6 demonstrates that velocity doesn't increase at a constant rate as distance increases. For example, between 0 to 1 foot, the increase in velocity is 8 ft/sec, while from 3 to 4 feet, it's only 2.1 ft/sec. This finding suggests that the relationship between distance fallen and velocity is more complex than a straightforward linear increase inside Galileo's hypothesis.

Experimental Data Analysis

Analyzing experimental data is crucial to validating or refuting scientific hypotheses. Step by step, Galileo's initial hypothesis was put to the test through observation and measurement. By analyzing data, like the velocity concerning distance in Table 1.6, we noticed inconsistencies with Galileo's expectations.
Data analysis involves calculating changes and spotting patterns; it's what turned Galileo away from the original notion of velocity and distance. Instead of finding constant changes in velocity with distance, we observe variability. Tables 1.6 and 1.7 challenge and refine our understanding. Table 1.6 suggested the need for a new hypothesis, while Table 1.7 with consistently rising velocity over time gave a new perspective. Understanding how experimental data analysis works illuminates how earlier scientific mistakes were corrected.

Uniform Acceleration

Uniform acceleration refers to a constant acceleration experienced by an object over time, regardless of other factors such as the distance traveled. When considering falling objects, Galileo eventually realized that gravity provides uniform acceleration to those objects. This means the object's velocity increases uniformly as time progresses—specifically, at approximately 9.81 m/s² when considering free fall near Earth.
In the case of our data from Table 1.7, every second saw an equal increase in velocity by 32 ft/sec, contrary to the distance-dependent hypothesis. This illustrates uniform acceleration readily. Instead of velocity increasing due to the distance object fell, it increased solely because of the time it had been falling. So when thinking of an apple falling from a tree, it isn't the height that determines how fast it goes but the time it keeps falling, all of which underscores the power of uniform acceleration in telling time's precise relationship to motion.