Irodov 1.180: Constant Acceleration Sets an Exponential Mass

Problem Irodov 1.180

A rocket of mass $m_0$ at $t=0$ moves with a constant acceleration of magnitude $w$ , directed along its own axis of motion. No external force acts on it. It ejects gas continuously, always at the constant speed $u$ relative to the rocket, straight backward. Find how the rocket’s mass $m$ varies with time $t$ .

A snapshot of the rocket at an arbitrary instant: current mass m, ejecting an element dm backward at speed u relative to the body. The forward arrow marks the axis, along which both the velocity and the constant acceleration w lie. — A snapshot of the rocket at an arbitrary instant: current mass $m$ , ejecting an element $dm$ backward at speed $u$ relative to the body. The forward arrow marks the axis, along which both the velocity and the constant acceleration $\vect w$ lie.

Given

$w$
$u$
$m(0)=m_0$

Find

m(t)

Solution

Step 1The variable-mass equation along the axis

Newton’s second law for a variable-mass body with no external force, projected on the axis of constant acceleration, relates the mass loss rate directly to $w$ .

m\frac{dv}{dt} = -u\frac{dm}{dt} \tag{1}

Since $dv/dt=w$ is constant by hypothesis, this is a first-order separable equation in $m$ alone.

mw = -u\frac{dm}{dt} \;\Longrightarrow\; \frac{dm}{m} = -\frac{w}{u}\,dt

Step 2Integrating from the initial mass

Integrating both sides from $t=0$ (where $m=m_0$ ) to an arbitrary time $t$ (where the mass is $m$ ) gives a logarithm on the left and a linear term on the right.

\begin{align*}\int_{m_0}^{m}\frac{dm'}{m'} &= -\frac{w}{u}\int_0^t dt' \\ \ln\frac{m}{m_0} &= -\frac{w}{u}t\end{align*}

Exponentiating both sides isolates $m$ .

m = m_0\,e^{-wt/u}

(2)

Check

At $t=0$ the boxed formula gives $m=m_0e^0=m_0$ , matching the stated initial condition. Differentiating it, $dm/dt=-(w/u)m$ , and substituting back into (1) gives $m\,dv/dt=-u\,dm/dt=w m$ , so $dv/dt=w$ exactly, confirming the constant acceleration the mass law was built to produce.

Remark

The mass decays exponentially with a time constant $u/w$ : the same body ejecting its exhaust faster (larger $u$ ) sheds mass more slowly for the same acceleration, since each unit of ejected mass then carries more momentum and less of it is needed. This is the same exponential that governs any constant-fractional-rate process; here the constant fraction $w/u$ is fixed by requiring $w$ itself, not the thrust or the mass flow rate, to stay constant.