Determining the Constant of Integration

Oct 13, 2025 by thesweetdesigns.com 109 views

Introduction

In this article, we will delve into the motion of a particle traveling along the x-axis. The particle's velocity is given by the function $v(t) = t^{0.3} + 2 \sin(t+5)$ , where $t$ represents time. We are also provided with the initial position of the particle, which is $x = -2$ when $t = 2$ . Our goal is to determine the position, velocity, and acceleration of the particle at any given time $t$ .

Position, Velocity, and Acceleration

To analyze the particle's motion, we need to understand the relationships between position, velocity, and acceleration. The position of the particle at time $t$ is denoted by $x(t)$ , the velocity is denoted by $v(t)$ , and the acceleration is denoted by $a(t)$ . The acceleration is the derivative of the velocity with respect to time, and the velocity is the derivative of the position with respect to time.

Calculating the Position

To find the position of the particle at any time $t$ , we need to integrate the velocity function $v(t) = t^{0.3} + 2 \sin(t+5)$ with respect to time. This can be done using the fundamental theorem of calculus, which states that the integral of a function is equal to the antiderivative of the function.

x(t) = \int v(t) dt = \int (t^{0.3} + 2 \sin(t+5)) dt

Using the power rule of integration and the trigonometric substitution, we can evaluate the integral:

x(t) = \frac{t^{1.3}}{1.3} - 2 \cos(t+5) + C

where $C$ is the constant of integration. We can determine the value of $C$ by using the initial condition $x = -2$ when $t = 2$ .

Determining the Constant of Integration

We are given that the position of the particle is $x = -2$ when $t = 2$ . We can use this information to determine the value of the constant of integration $C$ .

-2 = \frac{2^{1.3}}{1.3} - 2 \cos(2+5) + C

Simplifying the expression, we get:

-2 = \frac{2^{1.3}}{1.3} - 2 \cos(7) + C

Using a calculator to evaluate the expression, we get:

-2 = 1.419 - 2 \cos(7) + C

Now, we need to find the value of $\cos(7)$ . Using a calculator, we get:

\cos(7) = -0.993

Substituting this value into the expression, we get:

-2 = 1.419 + 1.986 + C

Simplifying the expression, we get:

-2 = 3.405 + C

Now, we can solve for $C$ :

C = -3.405 - 2

Simplifying the expression, we get:

C = -5.405

Position Function

Now that we have determined the value of the constant of integration $C$ , we can write the position function:

x(t) = \frac{t^{1.3}}{1.3} - 2 \cos(t+5) - 5.405

Velocity and Acceleration

To find the velocity and acceleration of the particle, we need to take the derivatives of the position function with respect to time.

v(t) = \frac{dx}{dt} = \frac{d}{dt} \left( \frac{t^{1.3}}{1.3} - 2 \cos(t+5) - 5.405 \right)

Using the power rule of differentiation and the chain rule, we get:

v(t) = t^{0.3} + 2 \sin(t+5)

This is the same as the given velocity function, which confirms that our position function is correct.

Acceleration

To find the acceleration of the particle, we need to take the derivative of the velocity function with respect to time.

a(t) = \frac{dv}{dt} = \frac{d}{dt} \left( t^{0.3} + 2 \sin(t+5) \right)

Using the power rule of differentiation and the chain rule, we get:

a(t) = 0.3 t^{-0.7} + 2 \cos(t+5)

Conclusion

In this article, we have analyzed the motion of a particle traveling along the x-axis. We have determined the position, velocity, and acceleration of the particle at any given time $t$ . The position function is given by:

x(t) = \frac{t^{1.3}}{1.3} - 2 \cos(t+5) - 5.405

The velocity function is given by:

v(t) = t^{0.3} + 2 \sin(t+5)

The acceleration function is given by:

a(t) = 0.3 t^{-0.7} + 2 \cos(t+5)

These functions can be used to analyze the particle's motion and make predictions about its future behavior.

Introduction

In our previous article, we analyzed the motion of a particle traveling along the x-axis. We determined the position, velocity, and acceleration of the particle at any given time $t$ . In this article, we will answer some frequently asked questions about the particle's motion.

Q: What is the initial position of the particle?

A: The initial position of the particle is $x = -2$ when $t = 2$ .

Q: What is the velocity of the particle at time $t$ ?

A: The velocity of the particle at time $t$ is given by the function $v(t) = t^{0.3} + 2 \sin(t+5)$ .

Q: What is the acceleration of the particle at time $t$ ?

A: The acceleration of the particle at time $t$ is given by the function $a(t) = 0.3 t^{-0.7} + 2 \cos(t+5)$ .

Q: How can I use the position function to determine the particle's position at any time $t$ ?

A: To determine the particle's position at any time $t$ , you can use the position function:

x(t) = \frac{t^{1.3}}{1.3} - 2 \cos(t+5) - 5.405

Q: How can I use the velocity function to determine the particle's velocity at any time $t$ ?

A: To determine the particle's velocity at any time $t$ , you can use the velocity function:

v(t) = t^{0.3} + 2 \sin(t+5)

Q: How can I use the acceleration function to determine the particle's acceleration at any time $t$ ?

A: To determine the particle's acceleration at any time $t$ , you can use the acceleration function:

a(t) = 0.3 t^{-0.7} + 2 \cos(t+5)

Q: What is the relationship between the position, velocity, and acceleration of the particle?

A: The position, velocity, and acceleration of the particle are related by the following equations:

v(t) = \frac{dx}{dt}
a(t) = \frac{dv}{dt} = \frac{d^2x}{dt^2}

Q: How can I use the position, velocity, and acceleration functions to make predictions about the particle's future behavior?

A: You can use the position, velocity, and acceleration functions to make predictions about the particle's future behavior by analyzing the functions and using them to determine the particle's position, velocity, and acceleration at future times.

Q: What are some real-world applications of the particle's motion?

A: The particle's motion has many real-world applications, including:

Modeling the motion of objects in physics and engineering
Analyzing the behavior of complex systems in biology and chemistry
Developing new technologies and products in fields such as robotics and aerospace engineering

Conclusion

In this article, we have answered some frequently asked questions about the particle's motion along the x-axis. We have provided the position, velocity, and acceleration functions, and explained how to use them to make predictions about the particle's future behavior. We have also discussed some real-world applications of the particle's motion.