A Particle Moves Along The X X X -axis. The Velocity Of The Particle At Time T T T Is Given By V ( T ) = 4 T 4 + 1 V(t)=\frac{4}{t^4+1} V ( T ) = T 4 + 1 4 .If The Position Of The Particle Is X = 1 X=1 X = 1 When T = 2 T=2 T = 2 , What Is The Position Of The Particle When
Introduction
In this article, we will delve into the world of mathematical modeling, specifically focusing on the motion of a particle along the x-axis. The velocity of the particle is given by the function v(t) = 4/(t^4 + 1), where t represents time. We are also provided with the initial position of the particle, x = 1, when t = 2. Our goal is to determine the position of the particle at a later time, t = 5.
The Relationship Between Velocity and Position
To find the position of the particle at a given time, we need to understand the relationship between velocity and position. The velocity of an object is defined as the rate of change of its position with respect to time. Mathematically, this can be expressed as:
v(t) = dx/dt
where v(t) is the velocity at time t, and dx/dt represents the rate of change of the position x with respect to time t.
The Position Function
Given the velocity function v(t) = 4/(t^4 + 1), we can find the position function x(t) by integrating the velocity function with respect to time t. This is based on the fundamental theorem of calculus, which states that the derivative of the position function is equal to the velocity function.
x(t) = ∫v(t)dt = ∫(4/(t^4 + 1))dt
Evaluating the Integral
To evaluate the integral, we can use the substitution method. Let u = t^4 + 1, then du/dt = 4t^3. We can rewrite the integral as:
x(t) = ∫(4/(t^4 + 1))dt = ∫(1/u)du = ln|u| + C
Substituting Back
Now, we can substitute back u = t^4 + 1 to obtain the position function x(t) in terms of t:
x(t) = ln|t^4 + 1| + C
Applying the Initial Condition
We are given that the position of the particle is x = 1 when t = 2. We can use this initial condition to find the value of the constant C.
x(2) = 1 ln|2^4 + 1| + C = 1 ln|17| + C = 1
Solving for C
Now, we can solve for the value of C:
C = 1 - ln|17|
The Position Function
Now that we have found the value of C, we can write the position function x(t) as:
x(t) = ln|t^4 + 1| + 1 - ln|17|
Simplifying the Position Function
We can simplify the position function by combining the logarithmic terms:
x(t) = ln|(t^4 + 1)/17| + 1
Finding the Position at t = 5
Finally, we can find the position of the particle at t = 5 by substituting t = 5 into the position function:
x(5) = ln|(5^4 + 1)/17| + 1 = ln|(625 + 1)/17| + 1 = ln|626/17| + 1 = ln|36.70| + 1 = 3.60 + 1 = 4.60
Conclusion
In this article, we have explored the motion of a particle along the x-axis, given by the velocity function v(t) = 4/(t^4 + 1). We have used the initial condition x = 1 when t = 2 to find the position function x(t) and then used this function to determine the position of the particle at t = 5. The position of the particle at t = 5 is approximately 4.60.
References
- [1] Calculus, 3rd edition, Michael Spivak
- [2] Differential Equations and Dynamical Systems, 3rd edition, Lawrence Perko
- [3] Mathematical Methods for Physics and Engineering, 3rd edition, K. F. Riley, M. P. Hobson, and S. J. Bence
Introduction
In our previous article, we explored the motion of a particle along the x-axis, given by the velocity function v(t) = 4/(t^4 + 1). We used the initial condition x = 1 when t = 2 to find the position function x(t) and then used this function to determine the position of the particle at t = 5. In this article, we will answer some of the most frequently asked questions related to this topic.
Q&A
Q: What is the relationship between velocity and position?
A: The velocity of an object is defined as the rate of change of its position with respect to time. Mathematically, this can be expressed as:
v(t) = dx/dt
where v(t) is the velocity at time t, and dx/dt represents the rate of change of the position x with respect to time t.
Q: How do you find the position function x(t) from the velocity function v(t)?
A: To find the position function x(t), you need to integrate the velocity function v(t) with respect to time t. This is based on the fundamental theorem of calculus, which states that the derivative of the position function is equal to the velocity function.
x(t) = ∫v(t)dt
Q: What is the significance of the initial condition x = 1 when t = 2?
A: The initial condition x = 1 when t = 2 is used to find the value of the constant C in the position function x(t). This constant is necessary to determine the position of the particle at any given time.
Q: How do you find the value of the constant C?
A: To find the value of the constant C, you need to substitute the initial condition x = 1 when t = 2 into the position function x(t) and solve for C.
Q: What is the position function x(t) in terms of t?
A: The position function x(t) is given by:
x(t) = ln|(t^4 + 1)/17| + 1
Q: How do you find the position of the particle at t = 5?
A: To find the position of the particle at t = 5, you need to substitute t = 5 into the position function x(t) and evaluate the expression.
x(5) = ln|(5^4 + 1)/17| + 1 = ln|(625 + 1)/17| + 1 = ln|626/17| + 1 = ln|36.70| + 1 = 3.60 + 1 = 4.60
Q: What is the significance of the logarithmic term in the position function x(t)?
A: The logarithmic term in the position function x(t) represents the rate of change of the position with respect to time. It is a measure of how quickly the position of the particle changes over time.
Q: Can you provide more examples of position functions?
A: Yes, here are a few more examples of position functions:
- x(t) = ∫(2t^2 + 1)dt = t^3 + t + C
- x(t) = ∫(3t^2 - 2t + 1)dt = t^3 - t^2 + t + C
- x(t) = ∫(4t^3 - 3t^2 + 2t - 1)dt = t^4 - t^3 + t^2 - t + C
Conclusion
In this article, we have answered some of the most frequently asked questions related to the motion of a particle along the x-axis. We have explored the relationship between velocity and position, found the position function x(t) from the velocity function v(t), and used the initial condition x = 1 when t = 2 to determine the position of the particle at t = 5. We hope that this article has provided a better understanding of the mathematical concepts involved in this topic.
References
- [1] Calculus, 3rd edition, Michael Spivak
- [2] Differential Equations and Dynamical Systems, 3rd edition, Lawrence Perko
- [3] Mathematical Methods for Physics and Engineering, 3rd edition, K. F. Riley, M. P. Hobson, and S. J. Bence