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Question #43464

Difficulty: 8/10 | 🟢 Free

The Question

A deep-space probe of initial rest mass $M$ is equipped with two sensor panels: one perfectly absorbing and one perfectly reflecting. A logic circuit on the probe has two inputs, $A$ and $B$ . Input $A$ registers '1' if the probe's internal energy increases by exactly $U_0$ . Input $B$ registers '1' if the probe's final kinetic energy is exactly $K_0$ . A single laser pulse of unknown energy $E$ is fired at the probe, illuminating both panels simultaneously. The reflected portion of the pulse travels back along the incident line. For this specific pulse, the logic gate $Y = A \cdot B$ outputs '1'. Assuming the probe is initially at rest in free space, what is the initial energy $E$ of the laser pulse?

(Note: $c$ is the speed of light in vacuum. Do not assume $K_0$ or $U_0$ are small compared to $Mc^2$ ).

Options

A.
$\frac{1}{2} \left( U_0 + K_0 + \sqrt{K_0^2 + 2K_0(Mc^2 + U_0)} \right)$
B.
$\frac{1}{2} \left( U_0 + K_0 + \sqrt{K_0^2 + 2K_0 Mc^2} \right)$
C.
$\frac{1}{2} \left( U_0 + K_0 + c\sqrt{2MK_0} \right)$
D.
2(U_0 + K_0)

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Logic Gate Boolean Algebra|Radiation Pressure|Recoil of Atom

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of Boolean logic conditions with rigorous relativistic mechanics.

The primary trap exploits a common misconception where students fail to account for the recoil momentum of the system when calculating radiation pressure or energy transfer (leading to Option D if they assume equal splitting and ignore momentum).

A secondary, more advanced trap tests the mass-energy equivalence principle: students often forget that an increase in internal energy ( $U_0$ ) fundamentally increases the rest mass of the macroscopic object to $M + U_0/c^2$ .

Failing to update the rest mass in the relativistic momentum-energy relation leads directly to the incorrect Option B, while using a non-relativistic approximation leads to Option C.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Semiconductor Electronics: Materials, Devices, and Simple Circuits - Class 12
Concept	Logic Gate Boolean Algebra
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free