Problem 50 - University Calculus: Early Transcendentals

Question

Give the acceleration \(a=d^{2} s / d t^{2},\) initial velocity. and initial position of an object moving on a coordinate line. Find the object's position at time \(t\). $$a=\frac{9}{\pi^{2}} \cos \frac{3 t}{\pi}, \quad v(0)=0, \quad s(0)=-1$$

Step-by-Step Solution

Answer

The position function is \( s(t) = -\pi \cos \left(\frac{3t}{\pi}\right) + \pi - 1 \).

Step 1: Find Velocity Function

To find the velocity as a function of time, integrate the acceleration function. Given: \[ a = rac{9}{\pi^{2}} \cos \left(\frac{3t}{\pi}\right) \]Integrate with respect to \( t \):\[ v(t) = \int \frac{9}{\pi^{2}} \cos \left(\frac{3t}{\pi}\right) \, dt \]The integral of \( \cos(kx) \) is \( \frac{1}{k} \sin(kx) \), thus:\[ v(t) = \frac{9}{3} \sin \left(\frac{3t}{\pi}\right) + C_1 = 3 \sin \left(\frac{3t}{\pi}\right) + C_1 \]

Step 2: Apply Initial Condition for Velocity

Use the initial condition \( v(0) = 0 \) to find the constant \( C_1 \).Set \( t = 0 \):\[ v(0) = 3 \sin(0) + C_1 = 0 \]\[ C_1 = 0 \]Thus, the velocity function is:\[ v(t) = 3 \sin \left(\frac{3t}{\pi}\right) \]

Step 3: Find Position Function

To find the position, integrate the velocity function:\[ v(t) = 3 \sin \left(\frac{3t}{\pi}\right) \]Integrate with respect to \( t \):\[ s(t) = \int 3 \sin \left(\frac{3t}{\pi}\right) \, dt \]The integral of \( \sin(kx) \) is \( -\frac{1}{k} \cos(kx) \), thus:\[ s(t) = -\pi \cos \left(\frac{3t}{\pi}\right) + C_2 \]

Step 4: Apply Initial Condition for Position

Use the initial condition \( s(0) = -1 \) to find the constant \( C_2 \).Set \( t = 0 \):\[ s(0) = -\pi \cos(0) + C_2 = -1 \]\[ -\pi + C_2 = -1 \]\[ C_2 = \pi - 1 \]The position function is:\[ s(t) = -\pi \cos \left(\frac{3t}{\pi}\right) + \pi - 1 \]

Step 5: Conclusion: Object's Position at Time t

The object's position at time \( t \) is given by the function:\[ s(t) = -\pi \cos \left(\frac{3t}{\pi}\right) + \pi - 1 \]

Key Concepts

Integration

Integration is a fundamental tool in calculus, often used to find quantities like areas under curves or, in physics, to determine quantities like velocity and position when the acceleration is known. In this exercise, we're tasked with integrating an acceleration function to find the velocity and position of an object over time.

• For our problem, the acceleration function given is \( a = \frac{9}{\pi^2} \cos \left(\frac{3t}{\pi}\right) \). Integration is needed to transform this into a velocity function.
• By finding the antiderivative of the acceleration, we determine that the velocity function becomes \( v(t) = 3 \sin \left(\frac{3t}{\pi}\right) + C_1 \).
• Further integration of the velocity function yields the position function, represented by \( s(t) = -\pi \cos \left(\frac{3t}{\pi}\right) + C_2 \).

Each integration step adds a constant of integration - these constants are crucial as they incorporate initial conditions to appropriately adjust our functions.

Initial Conditions

Initial conditions are the specific values provided to us which establish the exact solution of differential equations. These values ensure our equations represent the real-world scenario or precise requirements. They provide crucial information to determine constants of integration. In this exercise:

• The initial velocity is given as \( v(0) = 0 \). This condition allows us to solve for the constant \( C_1 \) in the velocity function. After using the initial condition, we find that \( C_1 = 0 \).
• The initial position, or where the object is initially situated, is given by \( s(0) = -1 \). By applying this, we solve for \( C_2 \) in the position function to get \( C_2 = \pi - 1 \).

Incorporating initial conditions means that we are not only solving a differential equation in general terms but making it specific to an initial real-world scenario.

Trigonometric Integrals

Trigonometric integrals involve the integration of functions containing trigonometric terms, such as sine and cosine functions. These integrals are common in problems involving periodic motion or wave-like phenomena in physics.

• In our exercise, the acceleration is given as a cosine function, \( \cos \left(\frac{3t}{\pi}\right) \), which represents periodic motion. The integration of cosine gives us a sine function, specifically \( v(t) = 3 \sin \left(\frac{3t}{\pi}\right) \) for our velocity.
• Similarly, when integrating the sine function of our velocity to find position, we end up with a cosine function, giving \( s(t) = -\pi \cos \left(\frac{3t}{\pi}\right) + \pi - 1 \) as our final position function.

Understanding these trigonometric integrals is essential for transitioning between different representations of motion, especially when the motion is oscillatory or cyclic.