Problem 8 - Calculus

Question

At time \(t\) in seconds, a particle's distance \(s(t),\) in centimeters, from a point is given by \(s(t)=4+3 \sin t .\) What is the average velocity of the particle from \(t=\pi / 3\) to \(t=7 \pi / 3 ?\)

Step-by-Step Solution

Answer

The average velocity is 0 cm/s.

Step 1: Define Average Velocity

The formula for average velocity \( v_{avg} \) over a time interval \([t_1, t_2]\) is given by the change in position (or distance) divided by the change in time: \[ v_{avg} = \frac{s(t_2) - s(t_1)}{t_2 - t_1} \] where \( s(t) \) is the position as a function of time.

Step 2: Evaluate the Position at the Start Time

Substitute \( t = \pi/3 \) into the position function \( s(t) = 4 + 3 \sin t \) to find the position at the start time:\[ s(\pi/3) = 4 + 3 \sin(\pi/3) = 4 + 3 \times \frac{\sqrt{3}}{2} = 4 + \frac{3\sqrt{3}}{2} \]

Step 3: Evaluate the Position at the End Time

Substitute \( t = 7\pi/3 \) into the position function \( s(t) = 4 + 3 \sin t \) to find the position at the end time:\[ s(7\pi/3) = 4 + 3 \sin(7\pi/3) \]Since \( 7\pi/3 \) is equivalent to \( 2\pi + \pi/3 \), \( \sin(7\pi/3) = \sin(\pi/3) = \frac{\sqrt{3}}{2} \), so:\[ s(7\pi/3) = 4 + 3 \times \frac{\sqrt{3}}{2} = 4 + \frac{3\sqrt{3}}{2} \]

Step 4: Calculate the Change in Position

Find the difference in position between \( t = \pi/3 \) and \( t = 7\pi/3 \):\[ \Delta s = s(7\pi/3) - s(\pi/3) = \left(4 + \frac{3\sqrt{3}}{2}\right) - \left(4 + \frac{3\sqrt{3}}{2}\right) = 0 \]

Step 5: Calculate the Time Interval

Calculate the time interval over which the average velocity is calculated:\[ \Delta t = 7\pi/3 - \pi/3 = 6\pi/3 = 2\pi \]

Step 6: Compute the Average Velocity

Using the formula for average velocity, substitute \( \Delta s = 0 \) and \( \Delta t = 2\pi \) to find:\[ v_{avg} = \frac{\Delta s}{\Delta t} = \frac{0}{2\pi} = 0 \]

Key Concepts

Position Function

When you're trying to determine the motion of a particle, the position function is essential. In this scenario, the position function is given as \(s(t) = 4 + 3 \sin t\). Here, \(s(t)\) represents the position of the particle at a certain time \(t\). This function involves a constant \(4\) and a term that varies with time due to the sine function, \(3 \sin t\).
This position function tells us that:

• At time \(t = 0\), the position is \(s(0) = 4 + 3 \sin(0) = 4\) since \(\sin(0) = 0\).
• The term \(3 \sin t\) will oscillate between \(-3\) and \(3\), because the sine of any angle is between \(-1\) and \(1\).
• Overall, the position \(s(t)\) of the particle changes over time because of this fluctuating sine term.

Understanding how to read and interpret position functions is key to mastering motion analysis.

Trigonometric Functions

Trigonometric functions play a crucial role in describing oscillations and waves, which is essential in understanding the motion of particles. Here, the sine function comes into play through the term \(3 \sin t\) in the position function.
Key properties of the sine function include:

• It is periodic with a period of \(2\pi\), meaning \(\sin(t + 2\pi) = \sin t\).
• It oscillates between \(-1\) and \(1\).
• For this exercise, the sine function's value computes the 'wave' that affects the particle's position about a constant offset (\(4\) in this case).

Knowing the periodic nature of the sine function allows for simplifying calculations, such as recognizing that \(\sin(7\pi/3)\) simplifies to \(\sin(\pi/3)\). This recognition helps in analyzing the particle’s repetitive position over time.

Time Interval

The time interval is the duration over which observations are made or calculations are performed. For average velocity, it's essential to know this interval as it helps in dividing the change in position.
In this problem, the time interval is from \(t = \pi/3\) to \(t = 7\pi/3\). To find this interval, we subtract the start time from the end time:

• Start Time: \(t_1 = \pi/3\)
• End Time: \(t_2 = 7\pi/3\)
• The Time Interval, \(\Delta t = t_2 - t_1 = 7\pi/3 - \pi/3 = 6\pi/3 = 2\pi\).

This time interval is crucial for calculating average velocity, helping you understand how long the changes occurred over time.

Change in Position

The change in position, often denoted as \(\Delta s\), measures how much the position of a particle has shifted during a given time period. In this problem, it involves finding the difference in position at two different times.
To compute \(\Delta s\):

• Calculate the position at \(t = \pi/3\), \(s(\pi/3) = 4 + 3 \sin(\pi/3) = 4 + \frac{3\sqrt{3}}{2}\).
• Calculate the position at \(t = 7\pi/3\), using the fact that \(\sin(7\pi/3) = \sin(\pi/3)\), \(s(7\pi/3) = 4 + \frac{3\sqrt{3}}{2}\).
• The change in position is \(\Delta s = s(7\pi/3) - s(\pi/3) = 0\).

In this case, the change in position is zero, meaning the particle returns to its original position, demonstrating how periodic motion impacts particle paths.