As F⃗=0, from the equation of dynamics of a body with...

As $\vec{F}=0$, from the equation of dynamics of a body with variable mass; $$ m \frac{d \vec{v}}{d t}=\vec{u} \frac{d m}{d t} \text { or, } d \vec{v}=\vec{u} \frac{d m}{m} $$ Now $d \vec{v} \uparrow \downarrow \vec{u}$ and since $\vec{u} \perp \vec{v}$, we must have $|d \vec{v}|=v_{0} d \alpha$ (because $v_{0}$ is constant) where $d \alpha$ is the angle by which the spaceship turns in time $d t$. So, or,

As $\vec{F}=0$, from the equation of dynamics of a body with variable mass;
$$
m \frac{d \vec{v}}{d t}=\vec{u} \frac{d m}{d t} \text { or, } d \vec{v}=\vec{u} \frac{d m}{m}
$$
Now $d \vec{v} \uparrow \downarrow \vec{u}$ and since $\vec{u} \perp \vec{v}$, we must have $|d \vec{v}|=v_{0} d \alpha$ (because $v_{0}$ is constant) where $d \alpha$ is the angle by which the spaceship turns in time $d t$.
So,
or,

Key Concepts

Relative Velocity

This concept refers to the velocity of the ejected (or incoming) mass relative to the system itself. It is crucial in formulating the equations of motion for variable mass systems because the dynamics depend not only on the rate at which mass changes, but also on the velocity at which the mass is expelled or received relative to the moving body.

Infinitesimal Analysis and Angular Displacement

In dynamics, infinitesimal changes such as d?v and d? are used to explore the system's response over very small intervals of time. When the speed of the body remains constant, a small change in the direction of velocity is related to the small angle d?. This approach helps in analyzing the turning motion or the gradual change in the trajectory of the body, linking the differential change in velocity to angular displacement.

Variable Mass Dynamics

This concept deals with systems whose mass is not constant over time, such as rockets and other space vehicles undergoing fuel consumption. In such systems, the standard form of Newton’s second law is modified to account for the change in mass, necessitating the inclusion of terms involving the rate of mass change (dm/dt) in the dynamics equations.

Conservation of Momentum

Even when a system’s mass varies, the total momentum of the system (including both the body and the ejected or accumulated mass) must be conserved in the absence of external forces. This principle is the foundation for deriving the dynamic equations in variable mass scenarios, ensuring that momentum lost or gained by the system is balanced by the momentum of the expelled or acquired mass.

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