Question

A mass \(m\) is accelerated by a time-varying force \(\exp (-\beta t) v^{3}\), where \(v\) is its velocity. It also experiences a resistive force \(\eta v\), where \(\eta\) is a constant, owing to its motion through the air. The equation of motion of the mass is therefore $$ m \frac{d v}{d t}=\exp (-\beta t) v^{3}-\eta v. $$ Find an expression for the velocity \(v\) of the mass as a function of time, given that it has an initial velocity \(v_{0}\).

Short answer

The velocity \( v(t) \) as a function of time is given by solving the integral: \[ t = \int \frac{m}{v ( \exp(-\beta t) v^2 - \eta )} dv\] and applying the initial condition.

Step-by-step solution

- Setup the Differential Equation

The given equation is: \[ m \frac{dv}{dt} = \exp(-\beta t) v^3 - \eta v \] We need to rewrite this as: \[ m \frac{dv}{dt} = v ( \exp(-\beta t) v^2 - \eta ) \] This is a separable differential equation.

- Separate the Variables

Separate the variables to isolate all terms involving velocity on one side and all terms involving time on the other: \[ m \frac{1}{v ( \exp(-\beta t) v^2 - \eta )} dv = dt \]

- Integrate Both Sides

We need to integrate both sides to find the velocity as a function of time. First, simplify and perform partial fraction decomposition if necessary. Rewrite the right-hand side as: \[ m \int \frac{1}{v ( \exp(-\beta t) v^2 - \eta )} dv = \int dt \]Integrate both sides.

- Use the Initial Condition

Solve the integrals and use the initial condition that when \( t = 0 \, \Rightarrow \, v = v_0 \): \[ t = \int \frac{m}{v ( \exp(-\beta t) v^2 - \eta )} dv\] Apply limits to find the constant of integration.

- Solve for Velocity

Solve the resulting equation for velocity \( v(t) \). This includes expressing \( v \) in terms of both \( t \) and the constants \( m, \beta, \eta, \) and the initial velocity \( v_0 \).

Key concepts

Time-Varying Forces

In physics, a time-varying force is a force that changes over time, as opposed to being constant. In our given problem, the force acting on the mass is represented by \( \exp(-\beta t) v^3 \). Here, \( \beta \) is a constant and \( t \) represents time. The term \( \exp(-\beta t) \) indicates that the force diminishes exponentially over time. This type of force commonly appears in scenarios involving damping or decaying processes.
Understanding time-varying forces is crucial when studying dynamic systems, as they can significantly influence the behavior of the system over time. For instance:

• Forces that change with time can lead to varying velocities and subsequently affect the position of objects in motion.
• In the given exercise, the exponential decay of the force means that its impact on accelerating the mass decreases over time, making it challenging to predict the motion without solving the differential equation.

To handle such forces in differential equations, it's important to carefully separate variables and integrate, as shown in the solution steps. This approach will allow us to find how the velocity evolves over time.

Resistive Forces

Resistive forces are forces that oppose the motion of an object through a medium, like air or water. In this exercise, the resistive force is given by \( \eta v \), where \( \eta \) is a constant and \( v \) is the velocity of the mass. The resistive force increases linearly with velocity, often leading to a drag force that acts against the direction of motion.

• Understanding resistive forces is vital in the study of dynamics, as these forces can significantly affect the acceleration and deceleration of objects.
• In the differential equation \( m \frac{dv}{dt} = \exp(-\beta t) v^3 - \eta v \), the resistive force term \( \eta v \) counteracts the acceleration due to the time-varying force.

To solve the equation, we need to balance the time-varying accelerating force and the resistive force. This requires setting up the differential equation and correctly separating the variables for integration.
Once the opposing forces are mathematically balanced through integration, we can determine how the resistive force modifies the motion over time.

Initial Conditions

Initial conditions are critical in solving differential equations as they provide specific values to determine unique solutions. In the given problem, the initial condition is that the mass has an initial velocity \( v_0 \) when \( t = 0 \). This allows us to find the constant of integration and uniquely determine the velocity function \((v(t))\).

• Initial conditions help to customize the general solution of a differential equation to the specific scenario in question.
• They act as boundary values, ensuring that the solution satisfies the given physical situation at the start of observation.

In our exercise, using the initial condition \( t = 0, v = v_0 \), we integrate both sides of the equation, apply limits, and solve for \( v(t) \). This step is necessary to reveal how the velocity of the mass changes over time, considering both the exponentially decaying time-varying force and the linear resistive force.