A Particle Travels Along The \[$ X \$\]-axis Such That Its Velocity Is Given By \[$ V(t) = T^{0.9} - 3 \cos(t^2 + 1) \$\]. The Position Of The Particle Is \[$ X = 4 \$\] When \[$ T = 2 \$\]. Determine The Position,
Introduction
In physics and mathematics, understanding the motion of particles is crucial for various applications, including engineering, computer simulations, and scientific research. The position, velocity, and acceleration of a particle are fundamental concepts that help us describe its motion. In this article, we will explore how to determine the position of a particle given its velocity function and initial conditions. We will use the given velocity function { v(t) = t^{0.9} - 3 \cos(t^2 + 1) $}$ and the initial position { x = 4 $}$ at time { t = 2 $}$ to find the position of the particle at any time { t $}$.
Velocity and Position Functions
The velocity function { v(t) $}$ represents the rate of change of the particle's position with respect to time. To find the position function { x(t) $}$, we need to integrate the velocity function with respect to time. The position function is the antiderivative of the velocity function.
Integrating the Velocity Function
To find the position function { x(t) $}$, we need to integrate the velocity function { v(t) = t^{0.9} - 3 \cos(t^2 + 1) $}$ with respect to time { t $}$. We can use the following integral:
{ x(t) = \int v(t) dt = \int (t^{0.9} - 3 \cos(t^2 + 1)) dt $}$
Evaluating the Integral
To evaluate the integral, we can use the following steps:
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Integrate the first term: The first term is { t^{0.9} $}$. We can integrate this term using the power rule of integration, which states that { \int t^n dt = \frac{t^{n+1}}{n+1} + C $}$.
{ \int t^{0.9} dt = \frac{t^{1.9}}{1.9} + C $}$
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Integrate the second term: The second term is { -3 \cos(t^2 + 1) $}$. We can integrate this term using the substitution method. Let { u = t^2 + 1 $}$. Then, { du = 2t dt $}$.
{ \int -3 \cos(t^2 + 1) dt = -\frac{3}{2} \int \cos(u) du $}$
{ = -\frac{3}{2} \sin(u) + C $}$
{ = -\frac{3}{2} \sin(t^2 + 1) + C $}$
Combining the Results
Now, we can combine the results of the two integrals to find the position function { x(t) $}$:
{ x(t) = \frac{t^{1.9}}{1.9} - \frac{3}{2} \sin(t^2 + 1) + C $}$
Applying the Initial Condition
We are given that the position of the particle is { x = 4 $}$ when { t = 2 $}$. We can use this initial condition to find the value of the constant { C $}$.
{ 4 = \frac{2^{1.9}}{1.9} - \frac{3}{2} \sin(2^2 + 1) + C $}$
{ 4 = \frac{2^{1.9}}{1.9} - \frac{3}{2} \sin(5) + C $}$
{ C = 4 - \frac{2^{1.9}}{1.9} + \frac{3}{2} \sin(5) $}$
Finding the Position Function
Now that we have found the value of the constant { C $}$, we can substitute it into the position function { x(t) $}$:
{ x(t) = \frac{t^{1.9}}{1.9} - \frac{3}{2} \sin(t^2 + 1) + 4 - \frac{2^{1.9}}{1.9} + \frac{3}{2} \sin(5) $}$
{ x(t) = \frac{t^{1.9}}{1.9} - \frac{3}{2} \sin(t^2 + 1) + 4 - \frac{2^{1.9}}{1.9} + \frac{3}{2} \sin(5) $}$
Conclusion
In this article, we have determined the position function of a particle given its velocity function and initial conditions. We have used the given velocity function { v(t) = t^{0.9} - 3 \cos(t^2 + 1) $}$ and the initial position { x = 4 $}$ at time { t = 2 $}$ to find the position function { x(t) $}$. We have integrated the velocity function to find the position function and applied the initial condition to find the value of the constant { C $}$. The resulting position function is { x(t) = \frac{t^{1.9}}{1.9} - \frac{3}{2} \sin(t^2 + 1) + 4 - \frac{2^{1.9}}{1.9} + \frac{3}{2} \sin(5) $}$.
References
- [1] Calculus, 3rd edition, Michael Spivak
- [2] Calculus, 2nd edition, Michael Spivak
- [3] Calculus, 1st edition, Michael Spivak
Introduction
In our previous article, we explored how to determine the position of a particle given its velocity function and initial conditions. We used the given velocity function { v(t) = t^{0.9} - 3 \cos(t^2 + 1) $}$ and the initial position { x = 4 $}$ at time { t = 2 $}$ to find the position function { x(t) $}$. In this article, we will answer some frequently asked questions related to the topic.
Q&A
Q: What is the velocity function?
A: The velocity function { v(t) $}$ represents the rate of change of the particle's position with respect to time. It is given by { v(t) = t^{0.9} - 3 \cos(t^2 + 1) $}$.
Q: What is the position function?
A: The position function { x(t) $}$ is the antiderivative of the velocity function. It is given by { x(t) = \frac{t^{1.9}}{1.9} - \frac{3}{2} \sin(t^2 + 1) + 4 - \frac{2^{1.9}}{1.9} + \frac{3}{2} \sin(5) $}$.
Q: How do I find the position function?
A: To find the position function, you need to integrate the velocity function with respect to time. You can use the power rule of integration and the substitution method to evaluate the integral.
Q: What is the initial condition?
A: The initial condition is the position of the particle at a given time. In this case, the initial position is { x = 4 $}$ at time { t = 2 $}$.
Q: How do I apply the initial condition?
A: To apply the initial condition, you need to substitute the initial position into the position function and solve for the constant { C $}$.
Q: What is the value of the constant { C $}$?
A: The value of the constant { C $}$ is given by { C = 4 - \frac{2^{1.9}}{1.9} + \frac{3}{2} \sin(5) $}$.
Q: What is the resulting position function?
A: The resulting position function is { x(t) = \frac{t^{1.9}}{1.9} - \frac{3}{2} \sin(t^2 + 1) + 4 - \frac{2^{1.9}}{1.9} + \frac{3}{2} \sin(5) $}$.
Q: Can I use this method to find the position function for any velocity function?
A: Yes, you can use this method to find the position function for any velocity function. However, you need to make sure that the velocity function is continuous and differentiable.
Q: What are some common applications of this method?
A: This method has many applications in physics, engineering, and computer simulations. Some common applications include:
- Finding the position of a particle given its velocity function and initial conditions
- Modeling the motion of objects in physics and engineering
- Simulating the behavior of complex systems in computer simulations
Conclusion
In this article, we have answered some frequently asked questions related to determining the position of a particle given its velocity function and initial conditions. We have used the given velocity function { v(t) = t^{0.9} - 3 \cos(t^2 + 1) $}$ and the initial position { x = 4 $}$ at time { t = 2 $}$ to find the position function { x(t) $}$. We have also discussed some common applications of this method and provided some tips for using this method.
References
- [1] Calculus, 3rd edition, Michael Spivak
- [2] Calculus, 2nd edition, Michael Spivak
- [3] Calculus, 1st edition, Michael Spivak