Problem 100 - Thomas Calculus

Question

Particle acceleration A particle moves along the \(x\) -axis with velocity \(d x / d t=f(x)\) . Show that the particle's acceleration is \(f(x) f^{\prime}(x) .\)

Step-by-Step Solution

Answer

The particle's acceleration is \(f(x) f'(x)\).

Step 1: Express velocity function

Given that the velocity of the particle is expressed as \(v = \frac{dx}{dt} = f(x)\). This means that the velocity function depends on the position \(x\).

Step 2: Use chain rule to differentiate velocity

To find the acceleration, we need to find the derivative of the velocity with respect to time \(t\). Since \(v = f(x)\), we use the chain rule: \(\frac{dv}{dt} = \frac{d}{dt}[f(x)] = f'(x) \cdot \frac{dx}{dt}\).

Step 3: Substitute velocity back into derivative

From Step 1, we know that \(\frac{dx}{dt} = f(x)\). Substituting this back into our derivative, we get \(\frac{dv}{dt} = f'(x) \cdot f(x)\).

Step 4: Simplify the expression for acceleration

Thus, the acceleration \(a\) of the particle can be expressed as \(a = f(x) f'(x)\). This is the result we wanted to show.

Key Concepts

Chain Rule

The chain rule is an essential concept in calculus used to differentiate composite functions. Often, we encounter functions that are expressed in terms of another function. The chain rule helps us find the derivative of such functions in a straightforward manner.
When you have a function composed of an outer function and an inner function, the chain rule states that the derivative of the composite function is the derivative of the outer function with respect to the inner function, multiplied by the derivative of the inner function with respect to the independent variable.
In the context of the problem, to find the acceleration from velocity, which is expressed as a function of position, the chain rule allows us to differentiate the velocity function conveniently by connecting the changes in velocity to the changes in position. This relationship is given by the formula:

• \[\frac{dv}{dt} = \frac{d}{dt}[f(x)] = f'(x) \cdot \frac{dx}{dt}\]

Velocity Function

The velocity function describes the speed and direction of a particle as it moves along a path. In this problem, the velocity is given by the function \(v = \frac{dx}{dt} = f(x)\).
This expression implies that the particle's velocity is not constant but varies depending on its position \(x\). Thus, the velocity is a function of space rather than time directly. When we express velocity as \(f(x)\), it gives us a clear understanding that for each position \(x\) on the path, there is a corresponding velocity determined by the function \(f\).
Understanding the velocity function is crucial because it forms the basis for determining the particle's acceleration by utilizing the relationship between the position and the velocity through differentiation.

Differentiation

Differentiation is a fundamental method in calculus used to determine how a function changes at any point. It provides the derivative, which represents the rate of change or the slope of the function at a specific point.
In the context of the velocity function \(v = f(x)\), differentiation becomes a tool to find the acceleration of a particle. Since acceleration is the change in velocity with respect to time, differentiating the velocity function \(f(x)\) with respect to time gives us the desired rate of change.
Using the chain rule, we differentiate \(f(x)\) as follows:

• Find the derivative of \(f(x)\) with respect to \(x\), which is \(f'(x)\).
• Multiply \(f'(x)\) by the derivative of \(x\) with respect to time \(\frac{dx}{dt}\) (which is \(f(x)\)).

This process gives the expression for acceleration in terms of the velocity function and its derivative.

Acceleration Formula

The acceleration formula is a crucial aspect of motion in physics, describing how the velocity of a particle changes with time. Acceleration is the second derivative of position with respect to time or the first derivative of velocity with respect to time.
In the current scenario, the given acceleration is expressed as a product of two components:

• The velocity function \(f(x)\), and
• Its derivative \(f'(x)\).

This is represented by the formula:

• \[a = f(x) \cdot f'(x)\]

This results from the chain rule application, where the product of the rate of change of the velocity function with respect to \(x\) and the velocity function itself gives the rate of change of velocity with respect to time.
Understanding this formula helps us see how the changes in position and the corresponding changes in the velocity function influence the acceleration of the particle along its path.