Problem 37 - 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 \(s=10 \cos \pi t\) a. What is the cart's maximum speed? When is the cart moving that fast? Where is it then? What is the magnitude of the acceleration then? b. Where is the cart when the magnitude of the acceleration is greatest? What is the cart's speed then?

Step-by-Step Solution

Answer

a. Max speed is \( 10\pi \) cm/s at \( t = \frac{1}{2} \), at position 0; acceleration is \( 0 \) there. b. Max acceleration magnitude is at \( \pm 10 \) cm, speed is 0 cm/s.

Step 1: Calculate Velocity Function

The position function is given as \( s(t) = 10 \cos(\pi t) \). To find the velocity, we differentiate \( s(t) \) with respect to time \( t \). The derivative of \( \cos(\pi t) \) is \( -\pi \sin(\pi t) \). Therefore, the velocity function is \( v(t) = -10\pi \sin(\pi t) \).

Step 2: Determine Maximum Speed

The maximum speed occurs when the magnitude of \( v(t) \) is at its peak, which is when \( \sin(\pi t) = \pm 1 \). The maximum speed is then \( |v(t)| = 10\pi \), at \( t = \frac{1}{2}, \frac{3}{2}, \ldots \).

Step 3: Find Position at Maximum Speed

At \( t = \frac{1}{2} \), the position is \( s(t) = 10 \cos\left(\pi \cdot \frac{1}{2}\right) = 10 \cdot 0 = 0 \). Therefore, the cart is at its equilibrium position, which is the rest position.

Step 4: Calculate Acceleration Function

To find the acceleration, differentiate the velocity function \( v(t) = -10\pi \sin(\pi t) \) with respect to \( t \). This yields \( a(t) = -10\pi^2 \cos(\pi t) \).

Step 5: Find Maximum Acceleration

The maximum magnitude of acceleration occurs when \( \cos(\pi t) = \pm 1 \). The acceleration is \( 10\pi^2 \), occurring at \( t = 0, 1, 2, \ldots \). At these points, the velocity \( v(t) = 0 \), meaning the cart momentarily stops at these extremes.

Step 6: Determine Position at Maximum Acceleration

At \( t = 0 \) or any integer \( t \), the position function \( s(t) = 10 \cos(\pi t) = \pm 10 \). Thus, the cart is at \( \pm 10 \) cm, furthest from equilibrium.

Key Concepts

Differentiation

Differentiation is a key concept in calculus that helps us determine the rate of change of a function. In simple terms, it tells us how a quantity changes as another quantity changes. In the context of our spring and cart system, the position of the cart, expressed as a function of time, is given by \( s(t) = 10 \cos(\pi t) \). Differentiation allows us to find how fast and in what direction the cart is moving—this is its velocity.

• Derivative: The derivative of a function represents its rate of change at any point. It is the foundation for finding both velocity and acceleration.
• Velocity Function: By differentiating the position function with respect to time \( t \), we obtain \( v(t) = -10\pi \sin(\pi t) \). It describes how quickly and in what direction the position changes.

Understanding differentiation not only helps in calculating velocity but also lays the groundwork for determining acceleration, which is the next level of change.

Velocity function

The velocity function tells us how the position of our cart changes over time. It gives both the speed (how fast) and the direction (which way) the cart is moving. In the exercise, the velocity function is derived from the position function by differentiation:

• Function details: The velocity function is \( v(t) = -10\pi \sin(\pi t) \). This indicates that the velocity is a function of the sine of \( t \), scaled by \(-10\pi\).
• Maximum velocity: The cart reaches its maximum speed when the absolute value of the sine function is 1, i.e., \( \sin(\pi t) = \pm 1 \). This occurs at times like \( t = \frac{1}{2}, \frac{3}{2} \), where the velocity \(|v(t)| = 10\pi\).

The velocity function is crucial in determining how the cart's motion changes over time and helps find moments of maximum speed and direction reversal.

Acceleration function

Acceleration is the rate of change of velocity. If you want to understand how quickly the cart's speed is changing, you need the acceleration function. This tells us how the velocity is adjusting itself in time. To find it, we differentiate the velocity function.

• Acceleration Formula: The exercise gives us the velocity function \( v(t) = -10\pi \sin(\pi t) \). Differentiating again, we get the acceleration: \( a(t) = -10\pi^2 \cos(\pi t) \).
• Maximum Acceleration: When the cosine function reaches \( \pm 1 \), the magnitude of acceleration is highest, resulting in \( |a(t)| = 10\pi^2 \). This happens at times like \( t = 0, 1, 2, \ldots \).
• Significance: When acceleration is at its maximum, the velocity is zero because the cart is changing direction. At those moments, the cart stops briefly before moving in the opposite direction.

Understanding acceleration allows us to predict how the motion of the cart evolves, especially at crucial extreme points where it changes direction.