Finding the Antiderivative

Oct 13, 2025 by leolavanderia.com.br 52 views

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Introduction

In this article, we will explore the motion of a particle moving along a line. The velocity of the particle at any time $t \geq 0$ is given by the function $v(t)=\frac{t}{t^2+2}$ . We are also given that at time $t=0$ , the position of the particle is $s(0)=7$ . Our goal is to find the position of the particle at time $t=4$ .

Velocity and Position

The velocity of an object is the rate of change of its position with respect to time. Mathematically, it is represented as $v(t)=\frac{ds}{dt}$ . In this case, the velocity function is given as $v(t)=\frac{t}{t^2+2}$ . To find the position of the particle at time $t=4$ , we need to find the antiderivative of the velocity function, which represents the position function $s(t)$ .

Finding the Antiderivative

To find the antiderivative of the velocity function, we can use the power rule of integration, which states that $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ . In this case, we have $v(t)=\frac{t}{t^2+2}$ , which can be rewritten as $v(t)=\frac{1}{t+2} \cdot t$ . Using the power rule, we can integrate this expression as follows:

\int v(t) dt = \int \frac{1}{t+2} \cdot t dt

= \int \frac{t}{t+2} dt

= \int \frac{t+2-2}{t+2} dt

= \int 1 - \frac{2}{t+2} dt

= t - 2\ln|t+2| + C

where $C$ is the constant of integration.

Evaluating the Antiderivative

Now that we have found the antiderivative of the velocity function, we can evaluate it at time $t=4$ to find the position of the particle. We are also given that at time $t=0$ , the position of the particle is $s(0)=7$ . We can use this initial condition to find the value of the constant $C$ .

s(0) = 7 = 0 - 2\ln|0+2| + C

C = 7 + 2\ln|2|

Now that we have found the value of the constant $C$ , we can evaluate the antiderivative at time $t=4$ to find the position of the particle.

s(4) = 4 - 2\ln|4+2| + 7 + 2\ln|2|

= 11 - 2\ln|6| + 2\ln|2|

= 11 - 2\ln\left(\frac{6}{2}\right)

= 11 - 2\ln(3)

Conclusion

In this article, we have found the position of a particle at time $t=4$ given its velocity function and initial position. We have used the power rule of integration to find the antiderivative of the velocity function, which represents the position function. We have then evaluated the antiderivative at time $t=4$ to find the position of the particle. The final answer is $s(4) = 11 - 2\ln(3)$ .

Final Answer

The final answer is: $\boxed{11 - 2\ln(3)}$

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Introduction

In our previous article, we explored the motion of a particle moving along a line. The velocity of the particle at any time $t \geq 0$ is given by the function $v(t)=\frac{t}{t^2+2}$ . We also found the position of the particle at time $t=4$ using the antiderivative of the velocity function. In this article, we will answer some frequently asked questions related to the motion of the particle.

Q&A

Q: What is the velocity of the particle at time $t=0$ ?

A: The velocity of the particle at time $t=0$ is given by $v(0)=\frac{0}{0^2+2}=0$ . This means that the particle is at rest at time $t=0$ .

Q: What is the position of the particle at time $t=0$ ?

A: The position of the particle at time $t=0$ is given as $s(0)=7$ . This is the initial position of the particle.

Q: How do we find the position of the particle at time $t=4$ ?

A: To find the position of the particle at time $t=4$ , we need to find the antiderivative of the velocity function, which represents the position function $s(t)$ . We can then evaluate the antiderivative at time $t=4$ to find the position of the particle.

Q: What is the relationship between velocity and position?

A: The velocity of an object is the rate of change of its position with respect to time. Mathematically, it is represented as $v(t)=\frac{ds}{dt}$ . This means that the velocity function is the derivative of the position function.

Q: Can we find the velocity function if we know the position function?

A: Yes, we can find the velocity function if we know the position function. We can take the derivative of the position function to find the velocity function.

Q: What is the significance of the constant $C$ in the antiderivative?

A: The constant $C$ in the antiderivative represents the initial position of the particle. We can use the initial condition to find the value of the constant $C$ .

Q: Can we find the position of the particle at any time $t$ ?

A: Yes, we can find the position of the particle at any time $t$ by evaluating the antiderivative at that time.

Conclusion

In this article, we have answered some frequently asked questions related to the motion of a particle. We have discussed the velocity and position of the particle, the relationship between velocity and position, and the significance of the constant $C$ in the antiderivative. We have also provided examples of how to find the position of the particle at any time $t$ .

Final Answer

The final answer is: $\boxed{11 - 2\ln(3)}$