Rocket Motion Suppose a small single-stage rocket of total...

Rocket Motion Suppose a small single-stage rocket of total mass $m(t)$ is launched vertically and that the rocket consumes its fuel at a constant rate. If the positive direction is upward and if we take air resistance to be linear, then a differential equation for its velocity $v(t)$ is given by $$ \frac{d v}{d t}+\frac{k-\lambda}{m_{0}-\lambda t} v=-g+\frac{R}{m_{0}-\lambda t^{\prime}} $$ where $k$ is the drag coefficient, $\lambda$ is the rate at which fuel is consumed, $R$ is the thrust of the rocket, $m_{0}$ is the total mass of the rocket at $t=0$, and $g$ is the acceleration due to gravity. See Problem 21 in Exercises $1.3$. (a) Find the velocity $v(t)$ of the rocket if $m_{0}=200 \mathrm{~kg}$, $R=2000 \mathrm{~N}, \lambda=1 \mathrm{~kg} / \mathrm{s}, g=9.8 \mathrm{~m} / \mathrm{s}^{2}, k=3 \mathrm{~kg} / \mathrm{s}$, and $v(0)=0$ (b) Use $d s / d t=v$ and the result in part (a) to find the height $s(t)$ of the rocket at time $t$.

Rocket Motion Suppose a small single-stage rocket of total mass $m(t)$ is launched vertically and that the rocket consumes its fuel at a constant rate. If the positive direction is upward and if we take air resistance to be linear, then a differential equation for its velocity $v(t)$ is given by
$$
\frac{d v}{d t}+\frac{k-\lambda}{m_{0}-\lambda t} v=-g+\frac{R}{m_{0}-\lambda t^{\prime}}
$$
where $k$ is the drag coefficient, $\lambda$ is the rate at which fuel is consumed, $R$ is the thrust of the rocket, $m_{0}$ is the total mass of the rocket at $t=0$, and $g$ is the acceleration due to gravity. See Problem 21 in Exercises $1.3$.
(a) Find the velocity $v(t)$ of the rocket if $m_{0}=200 \mathrm{~kg}$, $R=2000 \mathrm{~N}, \lambda=1 \mathrm{~kg} / \mathrm{s}, g=9.8 \mathrm{~m} / \mathrm{s}^{2}, k=3 \mathrm{~kg} / \mathrm{s}$, and
$v(0)=0$
(b) Use $d s / d t=v$ and the result in part (a) to find the height $s(t)$ of the rocket at time $t$.

Key Concepts

Rocket Thrust and Gravity

Rocket motion is influenced by competing forces: the upward thrust produced by the engine and the downward pull of gravity. The thrust provides an acceleration that is generally constant or dependent on time, while gravity exerts a constant acceleration downward. In the context of the differential equation, these forces are represented by constant or time-dependent terms that must be balanced to determine the rocket's overall motion.

Integrating Factor Method

This is a standard technique used to solve linear differential equations with variable coefficients. The method involves multiplying the entire differential equation by an integrating factor, which is designed so that the left-hand side of the equation becomes the derivative of a product of the integrating factor and the unknown function. This method is crucial for obtaining an explicit solution for the rocket's velocity over time.

Air Resistance Modeling

In many real-world dynamics problems, the drag force due to air resistance plays a significant role. When modeled as linear with respect to velocity, the drag force is represented as being proportional to the velocity. This linearity simplifies the analysis and leads to a term in the differential equation that competes with thrust and gravitational forces, affecting the rocket's acceleration.

Variable Mass Systems

This concept refers to physical systems where the mass changes over time, such as rockets that burn fuel. The dynamics of variable mass systems require special considerations because conventional laws of motion assume constant mass. In rocket propulsion, the decreasing mass affects acceleration and is typically modeled by incorporating the rate of mass change into the equations of motion.

First Order Linear Differential Equations

These equations, characterized by having the first derivative of a function along with the function itself multiplied by a time-dependent coefficient, are common in modeling dynamic systems. In the context of rocket motion with drag and variable mass, the velocity equation is a first order linear ODE that encapsulates the different forces acting on the rocket, such as thrust, gravity and air resistance.

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