Analyzing the Sinusoidal Term

Oct 13, 2025 by windailys.com 142 views

Introduction

In this article, we will delve into the motion of a particle moving along the x-axis. The velocity of the particle at any given time t is described by the function v(t) = 1 + 2 sin(t^2/2). We will explore the behavior of this function and determine the position of the particle at a specific time.

The Velocity Function

The velocity function v(t) = 1 + 2 sin(t^2/2) describes the rate of change of the particle's position with respect to time. The function consists of two parts: a constant term 1 and a sinusoidal term 2 sin(t^2/2). The constant term represents the particle's steady-state velocity, while the sinusoidal term represents the oscillatory motion of the particle.

Analyzing the Sinusoidal Term

The sinusoidal term 2 sin(t^2/2) is the key to understanding the particle's motion. This term oscillates between -2 and 2 as the argument t^2/2 varies. The frequency of the oscillation is determined by the coefficient of t^2 in the argument, which is 1/2. This means that the oscillation occurs at a rate that is proportional to the square root of time.

Evaluating the Velocity at Time t = 4

We are asked to evaluate the velocity at time t = 4. To do this, we substitute t = 4 into the velocity function:

v(4) = 1 + 2 sin(4^2/2) = 1 + 2 sin(8) = 1 + 2 sin(8)

Using a calculator or a trigonometric table, we find that sin(8) ≈ -0.1455. Therefore, the velocity at time t = 4 is:

v(4) ≈ 1 + 2(-0.1455) ≈ 1 - 0.291

The Position of the Particle at Time t = 4

To determine the position of the particle at time t = 4, we need to integrate the velocity function with respect to time. The position function x(t) is given by:

x(t) = ∫v(t) dt = ∫(1 + 2 sin(t^2/2)) dt

Evaluating the integral, we get:

x(t) = t + 2 ∫sin(t^2/2) dt

Using a trigonometric substitution, we can evaluate the integral:

x(t) = t + 2 ∫sin(u) du = t + 2(-cos(u)) + C = t - 2 cos(t^2/2) + C

where C is the constant of integration.

To determine the value of C, we use the initial condition x(0) = 0. Substituting t = 0 into the position function, we get:

x(0) = 0 - 2 cos(0^2/2) + C = 0 - 2 cos(0) + C = 0 - 2(1) + C = -2 + C

Therefore, the constant of integration C is equal to 2.

The Final Position of the Particle at Time t = 4

Now that we have determined the position function x(t), we can evaluate the position of the particle at time t = 4:

x(4) = 4 - 2 cos(4^2/2) = 4 - 2 cos(8) = 4 - 2 cos(8)

Using a calculator or a trigonometric table, we find that cos(8) ≈ 0.1455. Therefore, the position of the particle at time t = 4 is:

x(4) ≈ 4 - 2(0.1455) ≈ 4 - 0.291

Conclusion

In this article, we have analyzed the motion of a particle moving along the x-axis. The velocity of the particle at any given time t is described by the function v(t) = 1 + 2 sin(t^2/2). We have evaluated the velocity at time t = 4 and determined the position of the particle at that time. The results show that the particle's position is approximately 3.709 units from the origin at time t = 4.

References

[1] Calculus: Early Transcendentals, James Stewart, 8th edition.
[2] Trigonometry, Michael Corral, 2nd edition.

Appendix

The following is a list of formulas and theorems used in this article:

The derivative of the sine function: d(sin(x))/dx = cos(x)
The integral of the sine function: ∫sin(x) dx = -cos(x) + C
The trigonometric identity: sin^2(x) + cos^2(x) = 1
Q&A: Understanding the Motion of a Particle Along the x-axis ===========================================================

Frequently Asked Questions

In this article, we will address some of the most common questions related to the motion of a particle along the x-axis. We will provide detailed explanations and examples to help you better understand the concepts.

Q: What is the velocity function v(t) = 1 + 2 sin(t^2/2)?

A: The velocity function v(t) = 1 + 2 sin(t^2/2) describes the rate of change of the particle's position with respect to time. The function consists of two parts: a constant term 1 and a sinusoidal term 2 sin(t^2/2). The constant term represents the particle's steady-state velocity, while the sinusoidal term represents the oscillatory motion of the particle.

Q: What is the significance of the sinusoidal term 2 sin(t^2/2)?

A: The sinusoidal term 2 sin(t^2/2) is the key to understanding the particle's motion. This term oscillates between -2 and 2 as the argument t^2/2 varies. The frequency of the oscillation is determined by the coefficient of t^2 in the argument, which is 1/2. This means that the oscillation occurs at a rate that is proportional to the square root of time.

Q: How do you evaluate the velocity at time t = 4?

A: To evaluate the velocity at time t = 4, we substitute t = 4 into the velocity function:

v(4) = 1 + 2 sin(4^2/2) = 1 + 2 sin(8) = 1 + 2 sin(8)

Using a calculator or a trigonometric table, we find that sin(8) ≈ -0.1455. Therefore, the velocity at time t = 4 is:

v(4) ≈ 1 + 2(-0.1455) ≈ 1 - 0.291

Q: How do you determine the position of the particle at time t = 4?

A: To determine the position of the particle at time t = 4, we need to integrate the velocity function with respect to time. The position function x(t) is given by:

x(t) = ∫v(t) dt = ∫(1 + 2 sin(t^2/2)) dt

Evaluating the integral, we get:

x(t) = t + 2 ∫sin(t^2/2) dt

Using a trigonometric substitution, we can evaluate the integral:

x(t) = t + 2 ∫sin(u) du = t + 2(-cos(u)) + C = t - 2 cos(t^2/2) + C

where C is the constant of integration.

Q: What is the constant of integration C?

A: To determine the value of C, we use the initial condition x(0) = 0. Substituting t = 0 into the position function, we get:

x(0) = 0 - 2 cos(0^2/2) + C = 0 - 2 cos(0) + C = 0 - 2(1) + C = -2 + C

Therefore, the constant of integration C is equal to 2.

Q: What is the final position of the particle at time t = 4?

A: Now that we have determined the position function x(t), we can evaluate the position of the particle at time t = 4:

x(4) = 4 - 2 cos(4^2/2) = 4 - 2 cos(8) = 4 - 2 cos(8)

Using a calculator or a trigonometric table, we find that cos(8) ≈ 0.1455. Therefore, the position of the particle at time t = 4 is:

x(4) ≈ 4 - 2(0.1455) ≈ 4 - 0.291

Q: What is the significance of the particle's position at time t = 4?

A: The particle's position at time t = 4 is approximately 3.709 units from the origin. This means that the particle has moved 3.709 units from its initial position at time t = 0.

Q: What are some real-world applications of this concept?

A: This concept has many real-world applications, such as:

Modeling the motion of objects in physics and engineering
Understanding the behavior of complex systems in biology and chemistry
Developing algorithms for computer graphics and animation
Analyzing data in finance and economics

Conclusion

In this article, we have addressed some of the most common questions related to the motion of a particle along the x-axis. We have provided detailed explanations and examples to help you better understand the concepts. We hope that this article has been helpful in clarifying any doubts you may have had.