Thought Experiments and the History of the Law of Falling Bodies

Problem Statement

Person A decided to research the laws of falling objects and their history. Read the following passage and select the most appropriate number from the options that follow for each question.

“The ancient Greek $\fbox{1}$1​ was a great $\fbox{3}$3​ who lived more than $\fbox{2}$2​ years ago. He believed that when objects fall, ‘the heavier $\fbox{4}$4​ falls faster.’ This theory was widely accepted for a long time, but $\fbox{6}$6​, who was active in the $\fbox{5}$5​th century, exposed the contradiction in this theory through a thought experiment.

Suppose a heavy object and a light object are $\fbox{7}$7​ together with a string. If the old theory were correct, the slower-falling $\fbox{8}$8​ would hold the entire system back, so the two connected objects should fall at $\fbox{9}$9​. However, since connecting the two objects increases the total mass by $\fbox{10}$10​ compared to the original heavy object alone, according to the old theory, the connected objects should fall at $\fbox{11}$11​.

Thus, two contradictory conclusions are derived from the same premises. Therefore, we can conclude that the original theory is $\fbox{12}$12​. In reality, if $\fbox{13}$13​ is a $\fbox{14}$14​ space, then all objects are $\fbox{15}$15​.

(1) Select the person who fits $\fbox{1}$1​.

1: Isaac Newton 2: Galileo Galilei 3: Aristotle 4: Archimedes

(2) Select the most appropriate number of years, counting backward from the present year of 2026, that corresponds to $\fbox{2}$2​.

1: 500 2: 1100 3: 1700 4: 2300

(3) Select his primary title at that time to fill in $\fbox{3}$3​.

1: Physicist 2: Philosopher 3: Astronomer 4: Alchemist

(4) Select the term that fits in $\fbox{4}$4​.

1: Heavy 2: Light 3: Large in volume

(5) Select the century in which Galileo was active that fits in $\fbox{5}$5​.

1: 4 2: 12 3: 16 4: 20

(6) Select the person that fits in $\fbox{6}$6​.

1: Isaac Newton 2: Galileo Galilei 3: René Descartes 4: Copernicus

(7) Select the operation that fits in $\fbox{7}$7​.

1: Tied 2: Collided 3: Rubbed together

(8) Select the term that fits in $\fbox{8}$8​.

1: Heavy object 2: Light object 3: The string itself

(9) Select the description of the fall that fits in $\fbox{9}$9​.

1: A speed faster than that of a single heavy object 2: A speed between that of a heavy object and a light object 3: A speed slower than that of a light object alone

(10) Select the phrase that fits in $\fbox{10}$10​.

1: Heavy 2: Light 3: Unchanged

(11) Select the description of the fall that fits in $\fbox{11}$11​.

1: It must fall as slowly as possible 2: It must fall as fast as possible 3: It must come to a stop partway down

(12) Select the term that fits in $\fbox{12}$12​.

1: Completely correct 2: Incorrect 3: It’s hard to say

(13) Select the phenomenon that fits in $\fbox{13}$13​.

1: Air resistance 2: Gravitational force 3: Electrostatic force

(14) Select the phrase that fits in $\fbox{14}$14​.

1: If present (exists) 2: If absent (can be ignored)

(15) Select the law of falling objects that fits in $\fbox{15}$15​.

1: They fall simultaneously regardless of weight 2: They fall sequentially at speeds proportional to their weight 3: They fall at speeds inversely proportional to their volume

Constraints

• Let $A_x$Ax​ be the option number corresponding to the answer for each subquestion $(x)$(x).
• All $A_x$Ax​ values are natural numbers (1, 2, 3, or 4) indicating the correct answer for each option.

Input Format

Let $S_{\text{odd}} = A_1 + A_3 + A_5 + A_7 + A_9 + A_{11} + A_{13} + A_{15}$Sodd​=A1​+A3​+A5​+A7​+A9​+A11​+A13​+A15​ be the sum of the answers to the odd-numbered subproblems. Let $S_{\text{even}} = A_2 + A_4 + A_6 + A_8 + A_{10} + A_{12} + A_{14}$Seven​=A2​+A4​+A6​+A8​+A10​+A12​+A14​ be the sum of the answers to the even-numbered subproblems. Find the value of the product $S_{\text{odd}} \times S_{\text{even}}$Sodd​×Seven​ and enter that natural number.