Problem 11 - Calculus

Question

In a time of \(t\) seconds, a particle moves a distance of \(s\) meters from its starting point, where \(s=\sin (2 t)\) (a) Find the average velocity between \(t=1\) and \(t=\) \(1+h\) if: (i) \(\quad h=0.1\) (ii) \(\quad h=0.01,\) (iii) \(\quad h=0.001\) (b) Use your answers to part (a) to estimate the instantaneous velocity of the particle at time \(t=1\)

Step-by-Step Solution

Answer

The estimated instantaneous velocity at \( t = 1 \) is approximately \( -1.1 \).

Step 1: Understand the formula for average velocity

To compute the average velocity over a time interval, you use the formula:\[\text{Average Velocity} = \frac{s(t_2) - s(t_1)}{t_2 - t_1}\]In this problem, \( t_1 = 1 \) and \( t_2 = 1 + h \). The motion of the particle is described by the function \( s(t) = \sin(2t) \). Therefore, the average velocity is computed as follows:\[\text{Average Velocity} = \frac{\sin(2(1+h)) - \sin(2 \times 1)}{h}\]

Step 2: Calculate average velocity for each value of h

(i) For \( h = 0.1 \):\[\text{Average Velocity} = \frac{\sin(2 \times 1.1) - \sin(2 \times 1)}{0.1} = \frac{\sin(2.2) - \sin(2)}{0.1}\]Using a calculator:\[\text{Average Velocity} \approx \frac{0.808 - 0.909}{0.1} \approx -1.01\](ii) For \( h = 0.01 \):\[\text{Average Velocity} = \frac{\sin(2 \times 1.01) - \sin(2)}{0.01} = \frac{\sin(2.02) - \sin(2)}{0.01}\]Using a calculator:\[\text{Average Velocity} \approx \frac{0.897 - 0.909}{0.01} \approx -1.2\](iii) For \( h = 0.001 \):\[\text{Average Velocity} = \frac{\sin(2 \times 1.001) - \sin(2)}{0.001} = \frac{\sin(2.002) - \sin(2)}{0.001}\]Using a calculator:\[\text{Average Velocity} \approx \frac{0.9079 - 0.909}{0.001} \approx -1.1\]

Step 3: Estimate the instantaneous velocity as h approaches 0

The instantaneous velocity at \( t = 1 \) is the limit of the average velocity as \( h \) approaches 0. From our calculations, as \( h \) decreased, the average velocity converges closer to a consistent value. Therefore, we estimate the instantaneous velocity to be approximately:\[-1.1\]This estimation is reasonable based on the calculated average velocities for the smaller values of \( h \).

Key Concepts

Understanding Average Velocity

The concept of average velocity is crucial in understanding how an object's position changes over time. It represents the total displacement of an object divided by the total time taken. In mathematical terms, the average velocity over a time interval \(t_1\) to \(t_2\) can be expressed as:\[ \text{Average Velocity} = \frac{s(t_2) - s(t_1)}{t_2 - t_1} \]Here, \(s(t)\) is the position of the particle at time \(t\). By applying this formula to a function like \(s = \sin(2t)\), we can determine how the velocity changes over small time intervals.For example:

• When \( h = 0.1 \, 0.01 \, 0.001 \), the average velocity is given by evaluating the expression: \(\frac{\sin(2(1+h)) - \sin(2)}{h}\).

With each decreasing \(h\), the computed average velocities become more refined, approaching a final value, guiding us towards the instantaneous velocity.

Exploring the Sin Function

The sine function \(\sin(2t)\) is an essential part of trigonometry, and it provides a periodic wave-like behavior. This function helps us describe oscillatory or cyclic motions, which is often seen in physics and engineering.To break it down:

• The coefficient of \(t\) inside the sine function, here being \(2\), scales the input, affecting the wave's period.
• In general, \(\sin(x)\) oscillates between \(-1\) and \(1\).
• The function \(\sin(2t)\) specifically moves through its cycle more rapidly than \(\sin(t)\) since each value of \(t\) is multiplied by \(2\).

This rapid oscillation impacts velocity calculations, as observed through the shifting average velocities in our exercises. Understanding the behavior of the sine function is crucial for analyzing the particle's motion over time.

Comprehending the Limit of a Function

The concept of a limit is foundational in calculus, particularly in determining the instantaneous rate of change—what we often refer to as instantaneous velocity. The limit of a function as \(h\) approaches zero helps us find the derivative, which provides the instantaneous rate.Practically:

• It's used to refine an approximation by making the time interval infinitesimally small.
• When seeking the instantaneous velocity, we look at how the average velocity changes as \(h\) tends to \(0\).

For our specific example:\[ \lim_{{h \to 0}} \frac{\sin(2(1+h)) - \sin(2)}{h} \]This calculation shows us the rate of change of the sine function's output with respect to time, which is derived from the decreasing averages calculated. Thus, allowing us to conclude that the instantaneous velocity at \(t = 1\) is approximately \(-1.1\). The process of taking a limit refines our understanding beyond mere approximation to precise mathematical insight.