Rocket Velocity Optimization via Rocket Equation (Aerospace Engineering)

An aerospace propulsion engineer is analyzing the acceleration profile of a vertical launch vehicle during atmospheric flight. The burnout velocity $v$ (in meters per second, $\text{m/s}$) achieved by propellant consumption is governed by the Tsiolkovsky Rocket Equation:

where $v_e = 2500\text{ m/s}$ is the constant effective exhaust velocity, $m_0 = 50,000\text{ kg}$ is the initial fully fueled wet mass of the rocket, and $m$ is the instantaneous vehicle mass (in $\text{kg}$).

The rocket burns through its propellant fuel supply over time $t$ (in seconds) at a constant mass flow rate, meaning the instantaneous vehicle mass drops according to:

1. Pre-process the velocity equation using logarithmic properties to separate $\ln(m_0/m)$ into a difference of logarithms before differentiating.
2. Differentiate $v(m)$ with respect to mass $m$, and differentiate $m(t)$ with respect to time $t$.
3. Apply the Chain Rule in Leibniz notation ($\frac{dv}{dt} = \frac{dv}{dm} \cdot \frac{dm}{dt}$) to establish an analytical expression for the vehicle's acceleration $\frac{dv}{dt}$.
4. Evaluate the rocket's mass and instantaneous acceleration $\frac{dv}{dt}$ at exactly $t = 60\text{ seconds}$ into the burn sequence.