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Chapter 4: Problem 40

Give the velocity \(v=d s / d t\) and initial position of a body moving along a coordinate line. Find the body's position at time \(t\). \(v=\frac{2}{\pi} \cos \frac{2 t}{\pi}, \quad s\left(\pi^{2}\right)=1\)

Short Answer

Expert verified
The body's position at time \( t \) is \( s(t) = \sin \left(\frac{2t}{\pi}\right) + 1 \).

Step by step solution

01

Understand the Problem

The problem asks us to find the position function \( s(t) \) of a body given its velocity function and an initial condition. The velocity function is \( v(t) = \frac{2}{\pi} \cos \left(\frac{2t}{\pi}\right) \), and we know the initial position \( s(\pi^2) = 1 \).
02

Identify the Relation between Velocity and Position

The velocity is the derivative of the position function. To find the position function \( s(t) \), we need to integrate the velocity function \( v(t) = \frac{2}{\pi} \cos \left(\frac{2t}{\pi}\right) \).
03

Integrate the Velocity to Find Position

Integrate the velocity function:\[s(t) = \int \frac{2}{\pi} \cos \left(\frac{2t}{\pi}\right) dt\]This integration simplifies using the substitution method. Let \( u = \frac{2t}{\pi} \), then \( du = \frac{2}{\pi} dt \). It follows:\[\int \cos(u) \cdot \left(\frac{2}{\pi}\right) dt = \int \cos(u) \, du = \sin(u) + C\]Substituting back for \( u \), we have:\[s(t) = \sin \left(\frac{2t}{\pi}\right) + C\]
04

Apply the Initial Condition

Use the initial condition \( s(\pi^2) = 1 \) to find \( C \). Substitute \( t = \pi^2 \) into the position function:\[\sin \left(\frac{2\pi^2}{\pi}\right) + C = 1\]Simplifying, we get:\[\sin(2\pi) + C = 1\]Since \( \sin(2\pi) = 0 \), it follows that \( C = 1 \).
05

Write the Final Position Function

Substitute \( C = 1 \) back into the position function to obtain the final expression:\[s(t) = \sin \left(\frac{2t}{\pi}\right) + 1\]This is the position function for the body at any time \( t \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integration
Integration is the process we use to find the original function when we only know its derivative. In this exercise, the derivative we have is the velocity of a moving body, and we aim to find its position. The velocity function given is \( v(t) = \frac{2}{\pi} \cos \left(\frac{2t}{\pi}\right) \). By integrating this function, we acquire the position function \( s(t) \).

Here鈥檚 a breakdown of the integration process:
  • We start by identifying the integral that needs solving: \( s(t) = \int \frac{2}{\pi} \cos \left(\frac{2t}{\pi}\right) dt \).
  • Next, we use substitution鈥攁 common integration method. We substitute \( u = \frac{2t}{\pi} \), which simplifies the integral into a basic trigonometric integral: \( \int \cos(u) \cdot du = \sin(u) + C \).
  • Finally, we substitute back \( u \) to find the original function in terms of \( t \), resulting in: \( s(t) = \sin \left(\frac{2t}{\pi}\right) + C \).
Through integration, the trigonometric complexity is untangled, giving a clearer picture of the movement: the position as a function of time.
Initial Conditions
Initial conditions are crucial for determining the particular solution of an integrated function. They help us find the constant \( C \) of integration, which represents unknown starting values. In many real-world problems, physics especially, these conditions give us a reference point.

For this exercise, the initial condition is provided: \( s(\pi^2) = 1 \). This means that when time \( t \) is equal to \( \pi^2 \), the position \( s \) is exactly 1 unit. This value is pivotal for finding \( C \).
  • After integrating, you typically have a constant \( C \) to add. The initial condition allows us to solve for this constant.
  • By substituting \( t = \pi^2 \) into \( s(t) = \sin \left(\frac{2t}{\pi}\right) + C \), we use it to solve \( \sin(2\pi) + C = 1 \).
  • Knowing that \( \sin(2\pi) = 0 \), it becomes clear that \( C = 1 \).
Thus, applying initial conditions provided a tailor-made solution specific to the motion described.
Trigonometric Functions
Trigonometric functions are often used to model periodic phenomena, like waves or oscillations. In this problem, the velocity is expressed using a cosine function, \( \frac{2}{\pi} \cos \left(\frac{2t}{\pi}\right) \), suggesting a periodic movement. Integrating this yields a sine function due to the natural interplay between sine and cosine in calculus.
  • Cosine functions, like \( \cos \left(\frac{2t}{\pi}\right) \), are linked to repeated cycles, making them ideal to describe velocities over time.
  • When integrated, \( \cos(u) \) transforms into \( \sin(u) \), thus the position is modeled by \( \sin \left(\frac{2t}{\pi}\right) \).
  • The shift from cosine to sine upon integration reflects a phase change, capturing the transition from velocity to position.
By incorporating trigonometric functions, this solution captures the ebb and flow typical of periodic processes, hence offering a meaningful representation of movement.

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