Problem 49 - Thomas Calculus

Question

Frictionless cart A small frictionless cart, attached to the wall by a spring, is pulled 10 \(\mathrm{cm}\) from its rest position and released at time \(t=0\) to roll back and forth for 4 sec. Its position at time \(t\) is \(s=10 \cos \pi t.\) \begin{equation}\begin{array}{l}{\text { a. What is the cart's maximum speed? When is the cart moving }} \\ \quad {\text { that fast? Where is it then? What is the magnitude of the ac- }} \\ \quad {\text { celeration then? }} \\\ {\text { b. Where is the cart when the magnitude of the acceleration is }} \\\ \quad {\text { greatest? What is the cart's speed then? }}\end{array}\end{equation}

Step-by-Step Solution

Answer

Maximum speed is \(10\pi\), occurs at \(t = 0.5, 1.5, 2.5, 3.5\), position \(s = 0\). Max acceleration at rest positions.

Step 1: Understand the Problem

We have a cart attached to a spring moving back and forth, governed by the position function \( s(t) = 10 \cos{\pi t} \). We need to determine the maximum speed, the instances of maximum speed, the acceleration at that time, and find out when the acceleration is greatest.

Step 2: Find the Velocity Function

The velocity is the derivative of the position function. Calculate \( v(t) = \frac{d}{dt}[10 \cos{\pi t}] = -10 \pi \sin{\pi t} \).

Step 3: Determine the Maximum Speed

The speed is maximized when \( |v(t)| \) is maximized. Since \( v(t) = -10\pi \sin{\pi t} \), the maximum absolute value of \( \sin{\pi t} \) is 1. Therefore, the maximum speed is \( 10\pi \).

Step 4: Find When the Maximum Speed Occurs

\( \sin{\pi t} = \pm 1 \) when \( t = \frac{1}{2} + n \) for any integer \( n \). Considering the timeframe 0 to 4 seconds, \( t = 0.5, 1.5, 2.5, 3.5 \).

Step 5: Determine the Position at Maximum Speed

Substitute \( t = 0.5 \) (and similarly for others), into \( s(t) = 10 \cos{\pi t} \), which gives \( s(0.5) = 10 \cos{\frac{\pi}{2}} = 0 \).

Step 6: Calculate the Acceleration Function

The acceleration is the derivative of the velocity function: \( a(t) = \frac{d}{dt}[-10 \pi \sin{\pi t}] = -10 \pi^2 \cos{\pi t} \).

Step 7: Calculate Acceleration at Maximum Speed

At \( t = 0.5 \), \( a(0.5) = -10\pi^2 \cos{\pi/2} = 0 \). The acceleration is zero at maximum speed.

Step 8: Find Maximum Acceleration

Maximum magnitude of \( a(t) \) occurs when \( |\cos{\pi t}| = 1 \). This occurs when \( t = n \) for integer \( n \). In the given range, \( t = 0, 1, 2, 3, 4 \).

Step 9: Determine Position at Maximum Acceleration

If \( t = 0 \), for instance, \( s(0) = 10 \cos{0} = 10 \). Thus, the cart is at \( \pm 10 \) cm.

Step 10: Determine the Speed at Maximum Acceleration

At maximum acceleration, \( v(t) = 0 \) because the object is at rest at either turning point.

Key Concepts

Velocity Function

To understand harmonic motion, let's first look at the velocity function of the cart. The velocity function is derived from the position function by taking its derivative with respect to time. For the given problem:

• Position: \( s(t) = 10 \cos{\pi t} \)

The rate at which this position changes is the velocity:

• Velocity: \( v(t) = \frac{d}{dt}[10 \cos{\pi t}] = -10 \pi \sin{\pi t} \)

This function tells us that the velocity oscillates based on the sine function and is influenced by the amplitude of the initial displacement and the frequency determined by \( \pi \). Understanding this function helps predict how fast the cart will move at different points in time during its oscillation.

Acceleration Function

The acceleration function is all about understanding how the velocity changes over time. In this exercise, we determine the acceleration by differentiating the velocity function. Given:

• Velocity: \( v(t) = -10 \pi \sin{\pi t} \)

We find the acceleration:

• Acceleration: \( a(t) = \frac{d}{dt}[-10 \pi \sin{\pi t}] = -10 \pi^2 \cos{\pi t} \)

This acceleration function illustrates that the acceleration is also periodic and depends directly on the cosine function. The changes in acceleration indicate how quickly and in what manner the cart's speed is adjusting as it approaches different points of its oscillation. The acceleration is at maximum when the cart is at the extreme limits of its path, where the speed momentarily stops to change direction.

Maximum Speed

Achieving maximum speed in harmonic motion occurs when the velocity reaches its peak absolute value. For the cart:

• Velocity Function: \( v(t) = -10 \pi \sin{\pi t} \)

The sine function \( \sin{\pi t} \) can have values ranging from -1 to 1. Therefore, the maximum absolute value of velocity, and consequently the maximum speed, is:

• Maximum Speed: \( 10\pi \)

This occurs at times when \( \sin{\pi t} = \pm 1 \), specifically at \( t = 0.5, 1.5, 2.5, 3.5 \) seconds within the 4-second interval. At these moments, the cart is moving the fastest at the midpoint of its path, which is the equilibrium position.

Position Function

The position function is the starting point in solving problems related to harmonic motion, providing the displacement from rest over time. It is represented as:

• Position: \( s(t) = 10 \cos{\pi t} \)

This equation tells us that the cart's movement follows a cosine wave, starting from its maximum displacement of 10 cm. The cosine term indicates that the motion is periodic, meaning the cart returns to its starting point at regular intervals, creating a wave-like path as it moves back and forth along the track. This fundamental understanding of the position function helps in visualizing how the cart oscillates between the extremes of its path.