Problem 48 - Thomas Calculus

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 \(t .\) \begin{equation}a=\frac{9}{\pi^{2}} \cos \frac{3 t}{\pi}, \quad v(0)=0, \quad s(0)=-1\end{equation}

Step-by-Step Solution

Answer

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

Step 1: Integrate Acceleration to Find Velocity

The acceleration function is given by \( a(t) = \frac{9}{\pi^2} \cos \frac{3t}{\pi} \). To find the velocity function \( v(t) \), we need to integrate the acceleration function with respect to \( t \). Thus, \(\int a(t) \, dt = \int \frac{9}{\pi^2} \cos \frac{3t}{\pi} \, dt.\)Using the substitution \( u = \frac{3t}{\pi} \) gives \( du = \frac{3}{\pi} dt \) or \( dt = \frac{\pi}{3} du \). The integral becomes\( \frac{9}{\pi^2} \int \cos u \frac{\pi}{3} \, du = \frac{3}{\pi} \int \cos u \, du \).Evaluating the integral, \( \frac{3}{\pi} \sin u + C = \frac{3}{\pi}\sin\left(\frac{3t}{\pi}\right) + C \).So, \( v(t) = \frac{3}{\pi}\sin\left(\frac{3t}{\pi}\right) + C \).

Step 2: Use Initial Velocity Condition

We have the initial velocity condition \( v(0) = 0 \). Substitute \( t = 0 \) into the velocity equation:\[v(0) = \frac{3}{\pi} \sin\left(\frac{3 \cdot 0}{\pi}\right) + C = 0.\]Since \( \sin(0) = 0 \), we find that \( C = 0 \).Thus, the velocity function is:\[v(t) = \frac{3}{\pi}\sin\left(\frac{3t}{\pi}\right).\]

Step 3: Integrate Velocity to Find Position

To find the position function \( s(t) \), integrate the velocity function \( v(t) \). Thus,\[ s(t) = \int \frac{3}{\pi} \sin\left(\frac{3t}{\pi}\right) \, dt. \]Using the substitution \( u = \frac{3t}{\pi} \), \( du = \frac{3}{\pi} dt \), we have \( dt = \frac{\pi}{3} du \). Then the integral becomes\[\int \frac{3}{\pi} \sin u \frac{\pi}{3} \, du = \int \sin u \, du = -\cos u + C = -\cos\left(\frac{3t}{\pi}\right) + C.\]Therefore,\[ s(t) = -\cos\left(\frac{3t}{\pi}\right) + C.\]

Step 4: Use Initial Position Condition

The initial position condition is \( s(0) = -1 \). Substitute \( t = 0 \) into the position function to find \( C \):\[s(0) = -\cos\left(\frac{3 \cdot 0}{\pi}\right) + C = -1. \]\( \cos(0) = 1 \), so\[ -1 + C = -1, \] which gives \( C = 0 \).Then the position function is:\[ s(t) = -\cos\left(\frac{3t}{\pi}\right) \].

Step 5: Conclusion: Position at Any Time t

The position function describing the object's position over time is\[ s(t) = -\cos\left(\frac{3t}{\pi}\right). \]This position function is derived by integrating the given acceleration and applying initial conditions for velocity and position.

Key Concepts

Acceleration

Understanding acceleration is crucial in solving motion problems. Acceleration is the rate of change of velocity with respect to time. In simple terms, it tells us how quickly the velocity of an object is changing. For example, if an object's velocity is increasing, it has a positive acceleration; if decreasing, a negative acceleration. Acceleration is typically represented as \( a(t) = \frac{d^2 s}{dt^2} \), where \( s \) is the position function.
In our exercise, the acceleration function is given as \( a(t) = \frac{9}{\pi^2} \cos \left(\frac{3t}{\pi}\right) \). This function indicates that the acceleration is not constant but rather oscillates over time due to the cosine component.

• The presence of \( \cos \left(\frac{3t}{\pi}\right) \) means the acceleration follows a wave-like pattern.
• The constant \( \frac{9}{\pi^2} \) multiplies this wave, indicating the magnitude or amplitude of the acceleration.

Understanding this behavior helps us anticipate how the velocity and position of the object will change over time.

Velocity

Velocity represents an object's speed and direction. It is the first derivative of the position function with respect to time, indicating how the position changes over time. In mathematical terms, velocity \( v(t) \) is the integral of acceleration.
In this problem, to find the velocity function, we integrated the given acceleration, resulting in \( v(t) = \frac{3}{\pi} \sin \left(\frac{3t}{\pi}\right) + C \). After applying the initial condition \( v(0) = 0 \), we discovered \( C = 0 \), simplifying our velocity function to \( v(t) = \frac{3}{\pi} \sin \left(\frac{3t}{\pi}\right) \).

• This formula reveals that velocity also oscillates over time, with its behavior governed by a sine function.
• The constant \( \frac{3}{\pi} \) dictates the amplitude of these oscillations.
• The velocity changes as the argument of the sine function changes with time \( t \).

Thus, understanding velocity allows us to predict how fast and in what direction an object's position will change.

Position Function

The position function \( s(t) \) describes an object's specific location on a coordinate line at any given time. To find the position function, we integrate the velocity function.
In this particular exercise, after integrating the velocity function, we obtained \( s(t) = -\cos \left(\frac{3t}{\pi}\right) + C \). To find \( C \), we used the initial position condition \( s(0) = -1 \). This calculation showed that \( C = 0 \), which simplified the position function to \( s(t) = -\cos \left(\frac{3t}{\pi}\right) \).

• The \( \cos \left(\frac{3t}{\pi}\right) \) term implies that the position oscillates; it moves back and forth over time as a factor of cosine function.
• The negative sign in front signifies the direction of initial movement relative to the reference point.
• Evaluating this function at various times \( t \), allows predicting an object's precise location on its path.

By understanding the position function, we gain insight into how the initial conditions and time influence an object's trajectory.