Question

There are certain simple one-dimensional problems where the equation of motion (Newton's second law) can always be solved, or at least reduced to the problem of doing an integral. One of these (which we have met a couple of times in this chapter) is the motion of a one-dimensional particle subject to a force that depends only on the velocity \(v\), that is, \(F=F(v)\). Write down Newton's second law and separate the variables by rewriting it as \(m d v / F(v)=d t .\) Now integrate both sides of this equation and show that $$t=m \int_{v_{0}}^{v} \frac{d v^{\prime}}{F\left(v^{\prime}\right)}$$ Provided you can do the integral, this gives \(t\) as a function of \(v\). You can then solve to give \(v\) as a function of \(t .\) Use this method to solve the special case that \(F(v)=F_{\mathrm{o}},\) a constant, and comment on your result. This method of separation of variables is used again in Problems 2.8 and \(2.9 .\)

Short answer

For constant force, velocity is linear: \( v(t) = v_0 + \frac{F_0}{m}t \), showing constant acceleration motion.

Step-by-step solution

Write Newton's Second Law

Newton's second law for the motion of a particle is given by the equation \( F = ma \), where \( F \) is the force acting on the particle, \( m \) is its mass, and \( a \) is the acceleration. Acceleration \( a \) can be expressed as the derivative of velocity with respect to time \( a = \frac{dv}{dt} \). Therefore, we have \( F = m \frac{dv}{dt} \).

Substitute the Force Function

In this problem, the force \( F \) depends only on the velocity \( v \), i.e., \( F = F(v) \). Substitute this into Newton's second law equation to get \( F(v) = m \frac{dv}{dt} \).

Separate the Variables

Separate the variables by rearranging the equation \( F(v) = m \frac{dv}{dt} \) to solve for \( dt \): divide both sides by \( F(v) \) and multiply by \( dv \) to get \( \frac{m \, dv}{F(v)} = dt \). This allows us to integrate both sides with respect to their respective variables.

Integrate Both Sides

Integrate the left-hand side with respect to \( v \) from \( v_0 \) to \( v \), and the right-hand side with respect to \( t \) from 0 to \( t \). This gives the integral equation: \[ t = m \int_{v_0}^{v} \frac{dv'}{F(v')} \] Here, \( dv' \) indicates the variable of integration.

Solve for Constant Force Case

For the specific case where the force is constant, \( F(v) = F_0 \), the integral simplifies to: \[ t = m \int_{v_0}^{v} \frac{dv'}{F_0} = \frac{m}{F_0} \int_{v_0}^{v} dv' \] This straightforward integral results in \[ t = \frac{m}{F_0} (v - v_0) \].

Express Velocity as a Function of Time

Rearrange the result from the previous step to express velocity as a function of time: \( v - v_0 = \frac{F_0}{m} t \). Therefore, the velocity as a function of time is \( v(t) = v_0 + \frac{F_0}{m} t \), which represents linear motion with constant acceleration.

Key concepts

Velocity-Dependent Force

Velocity-dependent force means that the force acting on a particle is not constant, but rather depends on the velocity of the particle itself. This type of force is common in physics problems because it mimics real-world scenarios like drag or friction. For example, an object moving through a fluid experiences a drag force that depends on its speed. In the given problem, the force is expressed as a function of velocity, denoted as \( F(v) \). This setup requires a different approach than when dealing with constant forces.
The interesting aspect here is that velocity-dependent forces require us to rethink how we apply Newton's second law. Unlike constant forces, where the relationship between force, mass, and acceleration is straightforward, a change in velocity directly affects the force magnitude. Understanding this relationship is vital when analyzing bodies moving at varying speeds.
Newton's second law is given by \( F = ma \), where acceleration \( a \) can be replaced by the derivative of velocity \( a = \frac{dv}{dt} \). When the force depends on velocity, it becomes \( F(v) = m \frac{dv}{dt} \). This algebraic form shows us that the force taken as a function of velocity completely defines how the object's speed changes over time.

Separation of Variables

Separation of variables is a powerful mathematical technique often used to solve differential equations. It involves rearranging an equation so that each variable occupies a different side of the equation. In the context of Newton's second law for velocity-dependent forces, our goal was to isolate velocity on one side and time on the other.
Given the equation \( F(v) = m \frac{dv}{dt} \), separating the variables can be achieved by rearranging it to \( \frac{m \, dv}{F(v)} = dt \). This manipulation allows us to integrate both sides separately.

• On the left side, we integrate concerning the velocity, from the initial velocity \( v_0 \) to the current velocity \( v \).
• On the right side, the integration is in terms of time, from \( 0 \) to \( t \).

This results in the integral equation: \[t = m \int_{v_0}^{v} \frac{dv'}{F(v')}\]This equation lays the foundation for finding how a particle's velocity changes with time if starting from a known initial speed at \( t = 0 \).
For the specific case where \( F(v) = F_0 \), a constant force, separation of variables enables us to solve to find a simple linear solution, meaning time directly correlates with the change in velocity.

One-Dimensional Motion

One-dimensional motion simplifies analysis by reducing movement to a straight line, often examined along an axis. In physics, this constraint allows us to focus on how the magnitude of velocity changes along this single direction, free from complications of multi-dimensional systems.
When applying Newton's second law to one-dimensional motion, the resultant equations tend to be more straightforward. Consider a particle moving under the influence of a constant force \( F_0 \). Here, the relation becomes linear following the separations and integrations we've discussed.
In our example, the velocity of a particle, when subjected to a constant force, can be expressed by the equation \( v(t) = v_0 + \frac{F_0}{m} t \). This reveals the intuitive concept of linear acceleration and encapsulates the idea that in the absence of other directional forces, a particle will either speed up or slow down linearly depending on the sign and magnitude of force acting on it.
One-dimensional analysis underlines the basics of kinematics, allowing us to delve deeper into more complex systems once these principles are well understood. This fundamental approach is critical as it forms the basis for learning how objects move under various forces.