Problem 30 - Calculus Early Transcendentals: Pearson New International Edition

Question

At time \(t\) seconds, the center of a bobbing cork is \(3 \sin 2 t\) centimeters above (or below) water level. What is the velocity of the cork at \(t=0, \pi / 2, \pi\) ?

Step-by-Step Solution

Answer

Velocity at \( t=0 \) is 6 cm/s, at \( t=\frac{\pi}{2} \) is -6 cm/s, and at \( t=\pi \) is 6 cm/s.

Step 1: Understand the Function

The position of the cork is given by the function \( s(t) = 3 \sin 2t \). This function calculates the height of the cork above or below the water level at any time \( t \).

Step 2: Derive the Velocity Function

The velocity of the cork is the derivative of the position function. Using the chain rule, the derivative of \( s(t) = 3 \sin 2t \) is \( v(t) = \frac{d}{dt}[3 \sin 2t] = 3 \cdot 2 \cos 2t = 6 \cos 2t \).

Step 3: Calculate Velocity at Specific Times: \( t = 0 \)

Substitute \( t = 0 \) into the velocity function: \( v(0) = 6 \cos(2 \times 0) = 6 \cos(0) = 6 \times 1 = 6 \).

Step 4: Calculate Velocity at \( t = \frac{\pi}{2} \)

Substitute \( t = \frac{\pi}{2} \) into the velocity function: \( v\left(\frac{\pi}{2}\right) = 6 \cos\left(2 \times \frac{\pi}{2}\right) = 6 \cos(\pi) = 6 \times (-1) = -6 \).

Step 5: Calculate Velocity at \( t = \pi \)

Substitute \( t = \pi \) into the velocity function: \( v(\pi) = 6 \cos(2 \times \pi) = 6 \cos(2\pi) = 6 \times 1 = 6 \).

Step 6: Summarize Results

The velocity of the cork at \( t=0 \) is 6 cm/s, at \( t=\frac{\pi}{2} \) is -6 cm/s, and at \( t=\pi \) is 6 cm/s.

Key Concepts

chain rule

When differentiating a function that is composed of other functions, like our position function, the chain rule is a powerful tool. The chain rule allows us to differentiate composite functions by taking the derivative of the outer function and then multiplying it by the derivative of its inner function.

For the position function of the cork, given by (s(t) = 3 \sin 2t), the chain rule is applied by identifying the outer function as the sine function and the inner function as 2t.

Deriving the sine function gives \cos(2t), and multiplying by the derivative of the inner function, which is 2, results in the final velocity function \( v(t) = 6 \cos 2t \). This reveals how the cork's speed changes over time as it bobs up and down.

trigonometric functions

Trigonometric functions, like sine and cosine, are essential in calculus, especially when discussing periodic motion like that of a bobbing cork.

The sine function, \( \sin x \), is crucial in this problem because it models periodic movement, representing the motion above and below a central axis - in our case, the water level.

Understanding the properties of trigonometric functions is vital:

• They oscillate between -1 and 1, providing a natural model for repetitive motion.
• The derivative of sine is cosine, which is used to find velocity from position.
• Cosine, \( \cos x \), leads the sine function by a phase shift of 90 degrees, meaning its behavior is predictable next to sine.

These properties allow us to comprehend how changing time t shifts the cork's height and velocity as it moves periodically.

velocity calculation

Velocity represents the rate of change of position over time. To find the velocity of the cork, we calculate the derivative of the position function.

With our example, \( v(t) = 6 \cos 2t \), we evaluated this function at specific moments:

• At \( t=0 \), \( \cos(0) = 1 \), thus \( v(0) = 6 \).
• At \( t=\frac{\pi}{2} \), \( \cos(\pi) = -1 \), so \( v\left(\frac{\pi}{2}\right) = -6 \).
• At \( t=\pi \), \( \cos(2\pi) = 1 \), resulting in \( v(\pi) = 6 \).

These calculations emphasize how velocity can change sign and magnitude based on time, reflecting the periodic nature of the cork's movement. Positive values indicate upward motion, while negative values denote downward motion.

position function

A position function, like \( s(t) = 3 \sin 2t \), describes how an object's position changes over time.

This function uses sine to model periodic motion, such as the cork's bobbing, where:

• \( 3 \) is the amplitude, determining the extreme height or depth of the cork from the water surface.
• \( 2t \) indicates the frequency, controlling how often these peaks and troughs occur.

The position function not only provides the cork's vertical placement at any time t but also helps derive the velocity by differentiation.

In understanding the position function, one can visualize how the motion unfolds over time, anticipating the behavior of the cork as influenced by both amplitude and frequency.