Question

We need to design a ballistics chamber to decelerate test projectiles fired into it. Assume the resistive force encountered by the projectile is proportional to the square of its velocity and neglect gravity. As the figure indicates, the chamber is to be constructed so that the coefficient \(\kappa\) associated with this resistive force is not constant but is, in fact, a linearly increasing function of distance into the chamber. Let \(\kappa(x)=\kappa_{0} x\), where \(\kappa_{0}\) is a constant; the resistive force then has the form \(\kappa(x) v^{2}=\kappa_{0} x v^{2}\). If we use time \(t\) as the independent variable, Newton's law of motion leads us to the differential equation $$ m \frac{d v}{d t}+\kappa_{0} x v^{2}=0 \quad \text { with } \quad v=\frac{d x}{d t} . $$ (a) Adopt distance \(x\) into the chamber as the new independent variable and rewrite differential equation (14) as a first order equation in terms of the new independent variable. (b) Determine the value \(\kappa_{0}\) needed if the chamber is to reduce projectile velocity to \(1 \%\) of its incoming value within \(d\) units of distance.

Short answer

Answer: \(\kappa_{0} = \frac{2m(1 - 0.01)d^2}{0.01v_i^2}\).

Step-by-step solution

Step A: Rewrite the differential equation with distance x as the independent variable

First, we need to rewrite the given equation in terms of the distance x into the chamber. $$ m \frac{d v}{d t}+\kappa_{0} x v^{2}=0 \quad \text { with } \quad v=\frac{d x}{d t}. $$ First, we can use chain rule to rewrite \(\frac{dv}{dt}\) in terms of \(x\) as: $$ \frac{d v}{d t} = \frac{d v}{d x} \frac{d x}{d t} = v \frac{d v}{d x}. $$ Now, substitute this into the equation and separate variables: $$ m v \frac{d v}{d x}+\kappa_{0} x v^{2}=0. $$

Step B: Solve for the value of \(\kappa_{0}\) that reduces projectile velocity to \(1 \%\) within d units of distance

We are given that the chamber reduces the projectile's velocity to \(1\%\) of its incoming value within \(d\) units of distance. Let \(v_{i}\) and \(x_{i}\) be the initial velocity and initial distance, respectively. Then, at final distance \(d\), we have: $$ v_{f}=0.01 v_{i}. $$ We can now rewrite our equation in terms of \(x\) as follows: $$ m \int_{v_{i}}^{v_{f}} \frac{v}{v^2} dv = -\kappa_{0} \int_{x_{i}}^{d} x dx. $$ Solve the integrals: $$ -\frac{m}{v_i} + \frac{m}{0.01 v_i} = -\frac{\kappa_0}{2}d^2. $$ Now, we can simply solve for \(\kappa_{0}\): $$ \kappa_{0} = \frac{2m(1 - 0.01)d^2}{0.01v_i^2}. $$ Thus, the value of \(\kappa_{0}\) needed to reduce the projectile's velocity to \(1\%\) of its incoming value within \(d\) units of distance is: $$ \kappa_{0} = \frac{2m(1 - 0.01)d^2}{0.01v_i^2}. $$

Key concepts

Newton's Law of Motion

Newton's Law of Motion is fundamental to our understanding of physics. It describes how objects move in relation to the forces acting upon them. Simply put, Newton's second law of motion states that the force acting on an object is equal to the mass of that object multiplied by its acceleration, often expressed as \( F = ma \). In the context of our projectile problem, this law helps us create a differential equation to describe the motion of the projectile through the resistive medium.

By representing force as a function of velocity and position, we can derive a differential equation that denotes how the projectile slows down as it travels through the chamber. This equation is vital as it sets the groundwork for analyzing how various factors influence the projectile's deceleration. Understanding and applying Newton's laws allows us to model real-world ballistic scenarios effectively.

Resistive Force

Resistive Force plays a crucial role in how objects, like our projectile, interact with their surrounding environment. In the problem described, the resistive force is proportional to the square of the projectile's velocity \(v^2\) and acts in the opposite direction to the motion.

This type of resistive force often resembles air resistance or drag, which oppose the motion of an object moving at high speeds. The equation \(\kappa(x) v^{2}\), where \(\kappa(x)\) is a position-dependent coefficient, provides a mathematical model for this dynamic.

• It underscores how changes in the environment affect the deceleration rate.
• Helps simulate realistic projectile motion by accounting for resistive interactions.

Understanding resistive force helps us evaluate how fast a projectile will slow down and the conditions necessary for achieving specific outcomes within the chamber.

Ballistics

Ballistics refers to the science of the motion of projectiles and is pivotal to designing efficient systems for projectiles, such as the chamber discussed.

It encompasses various phenomena, including air resistance, gravity, and other forces acting on a projectile. Though gravity is neglected in the specific problem mentioned, ballistics traditionally considers all these factors.

• By focusing on resistive forces and velocity, the solution provides an insight into controlled deceleration without gravity interference.
• The resistive force's coefficient, \(\kappa(x)\), enhances the study of distance-based resistance in projectile motion.

Mastering ballistics helps physicists and engineers create environments where projectiles behave predictably, which is vital for both experimentation and practical applications.

Variable Coefficient Differential Equation

A Variable Coefficient Differential Equation, as seen here, contains terms where coefficients are not constant but depend on other variables, such as position \(x\).

In our ballistics chamber, the coefficient \(\kappa(x)\) changes with distance, represented by a linearly increasing function \(\kappa_{0}x\). This adjustment leads to new complexity in solving the differential equation, requiring techniques like separation of variables or integrating factors.

• These equations allow for more flexible modeling of dynamic systems, where conditions change over time or space.
• They also make the applied mathematics more reflective of real-world scenarios, especially in fluid dynamics and other physics applications.

Understanding how to set up and solve these equations is essential for anyone working with systems that cannot be described by constants alone.