Problem 41 - Calculus Early Transcendentals

Question

Prove that a particle is speeding up if the velocity and acceleration have the same sign, and slowing down if they have opposite signs. [Hint: Let \(r(t)=|v(t)|\) and find \(r^{\prime}(t)\) using the chain rule.]

Step-by-Step Solution

Answer

A particle speeds up if velocity and acceleration have the same sign, slows down if opposite signs.

Step 1: Understand the Problem

We are given a particle's velocity function \(v(t)\) and need to determine when the particle is speeding up or slowing down. The task is to use the velocity \(v(t)\) and the acceleration \(a(t) = v'(t)\) with the hint provided to analyze the speed \(r(t) = |v(t)|\).

Step 2: Define the Speed Function

The speed of the particle is the absolute value of the velocity, denoted as \(r(t) = |v(t)|\). This function represents the magnitude of the velocity, regardless of the direction.

Step 3: Differentiate the Speed Function

To find when the particle is speeding up or slowing down, we need to find the derivative of the speed, \(r'(t)\). Using the chain rule, we have: \[r'(t) = \frac{d}{dt}|v(t)| = \text{sign}(v(t)) \cdot v'(t) = \text{sign}(v(t)) \cdot a(t)\] where \(\text{sign}(v(t))\) is \(1\) if \(v(t) > 0\), \(-1\) if \(v(t) < 0\), and \(0\) if \(v(t) = 0\).

Step 4: Analyze the Result

If \(v(t) > 0\), then \(\text{sign}(v(t)) = 1\) and \(r'(t) = a(t)\). If \(a(t) > 0\), \(r'(t) > 0\), indicating the particle is speeding up. If \(a(t) < 0\), \(r'(t) < 0\), indicating the particle is slowing down. Similarly, if \(v(t) < 0\), then \(\text{sign}(v(t)) = -1\) and \(r'(t) = -a(t)\). If \(a(t) < 0\), \(r'(t) > 0\), indicating speeding up. If \(a(t) > 0\), \(r'(t) < 0\), indicating slowing down.

Step 5: Conclude the Analysis

The particle is speeding up when the signs of \(v(t)\) and \(a(t)\) are the same because \(r'(t) > 0\), increasing speed. It is slowing down when they have opposite signs because \(r'(t) < 0\), decreasing speed.

Key Concepts

The Basics of Velocity

When talking about particle motion, velocity is key. It tells us the speed and direction of a particle's motion at any given time. Velocity is a vector quantity, meaning it has both magnitude and direction. The function that gives us velocity is often denoted by \(v(t)\), with \(t\) being time. For example, if \(v(t) > 0\), it suggests the particle is moving in a positive direction, whereas \(v(t) < 0\) means it's moving negatively.

To understand when a particle speeds up or slows down, we have to consider not only the magnitude of velocity but also its direction:

• **Magnitude:** Refers to the absolute speed of the particle, without considering direction.
• **Direction:** Indicates which way the particle is moving along its path.

The absolute value \(|v(t)|\) gives us just the magnitude, which is what we refer to as the speed of the particle. This is crucial because it helps us focus on how fast the particle is moving, irrespective of its direction.

Acceleration and Its Role

Acceleration tells us how the velocity of a particle changes over time. It is represented by \(a(t)\) and is the derivative of the velocity function; mathematically, this is written as \(a(t) = v'(t)\). Acceleration affects both the speed and direction of motion:

• **Positive Acceleration:** If \(a(t) > 0\), the particle's velocity is increasing.
• **Negative Acceleration:** If \(a(t) < 0\), the particle's velocity is decreasing.

Acceleration can either support the current direction of movement or oppose it:

• When acceleration has the **same sign** as velocity, it boosts the speed of the particle. This means the particle is speeding up.
• If acceleration and velocity have **opposite signs**, the particle is resisting that motion, slowing down.

By analyzing acceleration in connection with velocity, we can comprehensively describe changes in a particle's motion.

Understanding the Chain Rule

The chain rule is a fundamental concept in calculus that helps us differentiate composite functions. It's essential when working with functions like velocity and acceleration in particle motion. Here's the basic idea:

• If you have a function \(f(g(x))\), the chain rule helps find the derivative as: \(f'(g(x)) \cdot g'(x)\).

In the context of particle motion, if we want to understand how the speed \(|v(t)|\) changes, we use the chain rule to differentiate \(|v(t)|\):

• This is done by recognizing that \(|v(t)|\) is a composite function, with the outside function being the absolute value and the inside being \(v(t)\).
• The chain rule gives us \(r'(t) = \text{sign}(v(t)) \cdot v'(t) = \text{sign}(v(t)) \cdot a(t)\).

This differentiation helps us determine the particle's motion more deeply, indicating when it accelerates or slows. Mastering the chain rule is pivotal for analyzing and interpreting real-world particle motion situations.