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Question #42360

Difficulty: 8/10 | 🟢 Free

The Question

A radioactive nucleus, initially at rest in a gravity-free space, undergoes continuous decay by emitting a stream of particles in a fixed direction. The speed of the emitted particles relative to the nucleus is a constant $u$ . The mass of the nucleus decreases according to the radioactive decay law $m(t) = m_0 e^{-\lambda t}$ , where $m_0$ is the initial mass and $\lambda$ is the decay constant. Which of the following expressions gives the distance $x(t)$ traveled by the nucleus as a function of time $t$ ?

Options

A.
$\frac{1}{2} u \lambda t^2$
B.
$u \left( \frac{e^{\lambda t} - 1}{\lambda} - t \right)$
C.
$u \left( t - \frac{1 - e^{-\lambda t}}{\lambda} \right)$
D.
$\frac{u}{\lambda} (1 - e^{-\lambda t})$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Variable Mass Systems|Recoil of Atom|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous application of Newton's Second Law for variable mass systems.

A common trap is for students to incorrectly apply macroscopic momentum conservation by assuming the emitted particles travel at speed $u$ in the lab frame, leading to $mv = (m_0 - m)u$ and an incorrect exponential distance function.

Another trap is assuming the thrust force is constant or failing to divide the instantaneous thrust by the instantaneous mass.

The problem elegantly demonstrates that an exponentially decreasing thrust applied to an exponentially decreasing mass results in a strictly constant acceleration, rewarding students who trust the differential formulation over intuitive guesses.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	System of Particles and Rotational Motion - Class 11 Part 1
Concept	Variable Mass Systems
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free