Pay-After-You-Play Amusement Park and the Digital Escape (AI) – Explanation
Problem Statement
Pay-After-You-Play Amusement Park and the Digital Escape (AI)
Problem Statement
"Light is not only a wave; it also possesses the properties of discrete packets of energy (particles)."
This concept, proposed by Albert Einstein, is known as the "light-quantum hypothesis." Light behaves as a collection of particles, each carrying an energy defined by E=hf (where h is Planck's constant and f is the frequency of the light).
When light with sufficient energy strikes the surface of a metal, the electrons within the metal absorb the energy from these light particles in their entirety and are ejected from the surface. This phenomenon is called the "photoelectric effect."
The process of an electron being ejected from the surface of a metal can be likened to a game: "Escaping from an amusement park that uses a 'pay-upon-exit' system."
- Exit Fee (Work Function, W) For a "guest" (an electron) inside the "amusement park" (the metal) to escape, they must pay an "exit fee"—an amount of energy (or money) equal to at least W—at the checkout counter near the exit.
- Found Banknote (Energy of a Single Light Particle, E=hf) Scattered on the ground inside the park are "banknotes"—light particles that have fallen from above—each with a value of E=hf. Each guest can pick up only one banknote; they cannot share a banknote with others, nor can they split it into smaller amounts.
- Remaining Funds upon Exit (Kinetic Energy, K)
- If the value of the banknote (E) is less than the required exit fee (W)—that is, E<W—the guest cannot afford the fee and, therefore, cannot escape the park. * If the value E of a banknote is greater than or equal to the admission fee W (E≥W), a visitor can pay the fee W at the exit counter to leave the park, with the remaining "change" (E−W) becoming their "exit momentum" (kinetic energy K).
In this scenario, the kinetic energy K (the change) of a visitor exiting the park is expressed by Einstein’s photoelectric equation:
K=E−W=hf−W
Suppose a visitor inside the park finds a banknote with a value of E1=hf1 (representing a light particle with frequency f1) and uses it to pay the admission fee; the visitor then exits the park with an amount of "change" (kinetic energy) equal to K1.
Next, if the value of the banknote found is increased fourfold to 4E1=h(4f1) (representing a light particle with a frequency four times greater), and a visitor uses this to pay the admission fee, the "change" (kinetic energy) they possess upon exiting becomes K2=13K1 (thirteen times the initial amount).
Determine the ratio of the admission fee (work function W) to the value of the first banknote found (E1=hf1).
Constraints
- f1>0Hz
- h>0J⋅s
- W>0J
Input Format
The ratio E1W can be expressed as an irreducible fraction qp (where p and q are coprime natural numbers). Provide the value of p×q.
Solution
Explanation
An Analogy: The Amusement Park and Einstein’s Photoelectric Equation
In the photoelectric effect, the energy E=hf of a single light particle (photon) with frequency f—analogous to the value of a banknote found on the ground—is transferred to a single electron within a metal.
If we define the work function W—the minimum energy an electron must "pay" to escape the metal—as the "admission fee" for an amusement park, then the maximum kinetic energy K of the ejected electron—analogous to the "change" received—is the energy transferred minus the admission fee.
K=hf−W=E−W
Formulating Equations for the Two Experimental Conditions
In the first experiment, when a "customer" finds a banknote with a value of E1=hf1, the "change" received is K1. Therefore, the following equation holds:
K1=E1−W— (1)
In the second experiment, the value of the banknote becomes 4E1 (four times the original value). In this case, the "change" received is 13K1 (13 times the initial amount K1). We can formulate the equation accordingly:
13K1=4E1−W— (2)
Calculating the Ratio Between the Admission Fee W and the Banknote Value E1
Multiply both sides of Equation (1) by 13:
13K1=13(E1−W)=13E1−13W
Since both this expression and Equation (2) represent the same value (13K1), they can be set equal to each other. 13E1−13W=4E1−W
Rearrange the equation by grouping E1 terms on the left side and W terms on the right side:
13E1−4E1=13W−W
9E1=12W
Divide both sides by 3:
3E1=4W
To find the required ratio E1W, divide both sides by 4E1:
E1W=43
Calculation of the Answer
The resulting ratio is E1W=43.
Here, the numerator p=3 and the denominator q=4; these are coprime natural numbers.
Following the specified input format, calculate p×q:
p×q=3×4=12
The final natural number to be entered is 12.