Problem 202 - Calculus Volume 1

Question

After a diver jumps off a diving board, the edge of the board oscillates with position given by \(s(t)=-5 \cos t \mathrm{cm}\) at \(t\) seconds after the jump. a. Sketch one period of the position function for \(t \geq 0\) . b. Find the velocity function. c. Sketch one period of the velocity function for \(t \geq 0\) . d. Determine the times when the velocity is 0 over one period. e. Find the acceleration function. f. Sketch one period of the acceleration function for \(t \geq 0\)

Step-by-Step Solution

Answer

a. Graph is a cosine wave from -5 to 5 from 0 to \(2\pi\). b. \(v(t)=5\sin t\). c. Sine wave from 0 to \(-5\) to 5. d. Velocity zero at \(t=0,\pi,2\pi\). e. \( a(t)=5\cos t\). f. Cosine wave from 5 to -5.

Step 1: Sketch Position Function

To sketch one period of the position function, start by understanding the function: \( s(t) = -5 \cos t \). This represents a cosine wave with amplitude 5, oscillating between -5 and 5. The negative sign reflects it over the horizontal axis. The period of the cosine function is \(2\pi\), so the graph will make a full cycle from \(t = 0\) to \(t = 2\pi\). Start at \(s(0) = -5\), graph passing through \(s(\pi/2) = 0\), \(s(\pi) = 5\), \(s(3\pi/2) = 0\), and \(s(2\pi) = -5\).

Step 2: Derive Velocity Function

Velocity is the derivative of position with respect to time. Given \(s(t) = -5\cos t\), the derivative is \(v(t) = \frac{ds}{dt} = 5\sin t\). The velocity function is \(v(t) = 5\sin t\).

Step 3: Sketch Velocity Function

The velocity function \(v(t) = 5\sin t\) oscillates with an amplitude of 5, and it follows a regular sine wave pattern. From \(t = 0\) to \(t = 2\pi\), the wave starts at 0, reaches maximum 5 at \(t = \pi/2\), goes through 0 at \(t = \pi\), reaches minimum -5 at \(t = 3\pi/2\), and returns to 0 at \(t = 2\pi\).

Step 4: Find Times When Velocity is Zero

The velocity is zero when \(v(t) = 5\sin t = 0\). This occurs when \(\sin t = 0\). For the interval from \(0\) to \(2\pi\), the solutions are \(t = 0, \pi, 2\pi\).

Step 5: Derive Acceleration Function

Acceleration is the derivative of velocity with respect to time. Start from \(v(t) = 5\sin t\), the derivative is \(a(t) = \frac{dv}{dt} = 5\cos t\). The acceleration function is \(a(t) = 5\cos t\).

Step 6: Sketch Acceleration Function

The acceleration function \(a(t) = 5\cos t\) has the same form as the position function but without a negative sign. It oscillates between -5 and 5 over a period \(2\pi\). Sketch starting at \(5\) at \(t = 0\), going through 0 at \(t = \pi/2\), reaching -5 at \(t = \pi\), returning to 0 at \(t = 3\pi/2\), and back to 5 at \(t = 2\pi\).

Key Concepts

Position Function

When discussing harmonic motion, the position function plays a pivotal role in understanding how objects move over time. In this case, the position function is given by \(s(t) = -5 \cos t\), where \(s(t)\) describes the position of the edge of the diving board at a specific time \(t\). This equation represents a cosine function characterized by:

• Amplitude: 5, which indicates the maximum displacement from the mean position (range is -5 to 5).
• Period: The time it takes to complete one full cycle, which for a cosine function is \(2\pi\).
• Reflection: The negative sign before the cosine flips the wave about the horizontal axis, starting at maximum (inverted).

At \(t = 0\), the position \(s(0)\) is at its lowest point, -5cm. Over a period of \(2\pi\), it oscillates from -5 to 5 and back. The simple oscillatory motion of the function helps predict future positions by relying on periodic behavior, showing how trigonometric functions describe wave-like motion.

Velocity Function

The velocity function is essential in understanding the speed and direction of motion over time. It is determined by deriving the position function with respect to time. From the position function, \(s(t) = -5 \cos t\), the velocity function becomes \(v(t) = 5 \sin t\).

• Amplitude: Like the sine function, it has an amplitude of 5, indicating the maximum speed.
• Sine Wave: The function follows the classic sine wave pattern, oscillating between -5 and 5.
• Zeros: The velocity is 0 when \(t = 0, \pi, 2\pi\), where the direction of motion momentarily pauses.

The velocity function captures the instantaneous rate of change of position. It tells us not only how fast the board is oscillating but also in which direction it’s moving at any given moment. Understanding the points where velocity equals zero can help in identifying moments of change in direction. This is vital in real-world applications of oscillatory motion, such as engineering and physics.

Acceleration Function

Acceleration provides insight into how velocity changes over time. To determine the acceleration function, we differentiate the velocity function \(v(t) = 5 \sin t\). This results in the acceleration function \(a(t) = 5 \cos t\).

• Amplitude: The amplitude of 5 shows the extent of change in velocity per unit time.
• Cosine Wave: The acceleration function mirrors the position function in shape, oscillating between -5 and 5.
• Phase Shift: Unlike position, the motion starts at peak acceleration when \(t = 0\).

In physical terms, acceleration describes how the board's speed increases or decreases over time. It shares the same frequency and amplitude as the position and velocity functions but leads them in phase. This alignment indicates the intricate relationship between these trigonometric functions. In oscillatory systems, understanding acceleration is critical for predicting behavior under various forces, making it a key component in dynamics.