Before You Begin

Question #42498

Difficulty: 8/10 | 🟢 Free

The Question

A rocket of initial mass $m_0$ is launched horizontally from rest in a viscous medium. The rocket ejects fuel backwards at a constant rate $r$ (in $\text{kg s}^{-1}$ ) with a constant relative exhaust velocity $u$ (in $\text{m s}^{-1}$ ). The viscous medium exerts a drag force $F_d = -kv$ , where $v$ is the instantaneous velocity of the rocket and $k$ is a positive constant. Assuming gravity is negligible, which of the following expressions correctly represents the velocity $v(t)$ of the rocket as a function of time $t$ (for $t < m_0/r$ )?

Options

A.
$\frac{ur}{k} \left[ 1 - \left( 1 - \frac{rt}{m_0} \right)^{\frac{k}{r}} \right]$
B.
$u \ln \left( \frac{m_0}{m_0 - rt} \right) - \frac{k}{m_0} t$
C.
$\frac{ur}{k} \left[ 1 - \exp\left(-\frac{kt}{m_0 - rt}\right) \right]$
D.
$\frac{ur}{k+r} \left[ 1 - \left( 1 - \frac{rt}{m_0} \right)^{\frac{k+r}{r}} \right]$

Check Your Answer

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

Recoil and Thrust Forces|Variable Mass Systems|Differential Equations

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous application of Newton's second law for variable mass systems, specifically $m(t)dv/dt = F_{ext} + F_{thrust}$ .

The primary trap is the presence of a velocity-dependent drag force alongside a time-varying mass, which creates a linear differential equation with variable coefficients.

Students commonly fail to recognize the need for an integrating factor, either defaulting to the standard Tsiolkovsky equation and subtracting a naive drag term, or treating the mass as constant during the integration of the drag force.

Solving this requires setting up the correct ODE and executing the integration with the proper boundary conditions.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Laws of Motion - Class 11 Part 1
Concept	Recoil and Thrust Forces
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free