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Question #42361

Difficulty: 8/10 | 🟢 Free

The Question

A spacecraft of initial mass $m_0$ is equipped with a perfectly absorbing solar sail of area $A$ . It is propelled by a constant radiation pressure $P$ from a distant star, which exerts a force $F = PA$ on the sail. The spacecraft travels along a straight line through a stationary dust cloud of uniform mass density $\rho$ . Assuming all dust striking the sail sticks to it and the spacecraft starts from rest at $t=0$ , which of the following expressions gives the velocity $v$ of the spacecraft as a function of time $t$ ?

Options

A.
$\frac{P A t}{\sqrt{m_0^2 + \rho A^2 P t^2}}$
B.
$\frac{P A t}{m_0 + t \sqrt{\rho A^2 P}}$
C.
$\frac{P A t}{\sqrt{m_0^2 + 2 \rho A^2 P t^2}}$
D.
$\frac{2 P A t}{\sqrt{m_0^2 + \rho A^2 P t^2}}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Variable Mass Systems|Radiation Pressure|Calculus

Why This Question Is Tricky

Common Mistakes

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

A common trap is the 'Reference Frame Pitfall' where students incorrectly apply $F = m(t)a$ without accounting for the momentum flux of the accreting mass.

Because the dust is stationary in the star's frame, the inelastic accretion acts as a drag force ( $v \frac{dm}{dt}$ ).

By using the momentum form $F = \frac{dp}{dt}$ , students bypass the complex non-linear differential equation $PA = m\frac{dv}{dt} + \rho A v^2$ and can directly integrate to find $p(t)$ , subsequently linking it to the spatial mass distribution $m(x)$ to solve for $v(t)$ .

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