The Position Of A Particle Moving Along A Coordinate Line Is S = 2 + 2 Π S=\sqrt{2+2 \pi} S = 2 + 2 Π ​ , With S S S In Meters And T T T In Seconds. Find The Particle's Velocity At T = 1 T = 1 T = 1 Sec.A) 1 4 M/sec \frac{1}{4} \, \text{m/sec} 4 1 ​ M/sec B)

Introduction

In physics, the position of a particle is often described by a function of time, known as the position function. This function, denoted by $s(t)$, gives the position of the particle at time $t$. In this article, we will explore the position function of a particle moving along a coordinate line, and use it to find the particle's velocity at a given time.

The Position Function

The position function of the particle is given by $s(t) = \sqrt{2+2 \pi}$. This function describes the position of the particle at time $t$, and is measured in meters.

Understanding the Position Function

To understand the position function, let's break it down into its components. The function $s(t) = \sqrt{2+2 \pi}$ is a square root function, which means that it takes the square root of the expression inside the parentheses. In this case, the expression inside the parentheses is $2+2 \pi$.

The Meaning of the Position Function

The position function $s(t) = \sqrt{2+2 \pi}$ tells us that the position of the particle at time $t$ is equal to the square root of $2+2 \pi$. This means that the position of the particle is constantly changing, and is dependent on the time $t$.

Finding the Velocity

To find the velocity of the particle, we need to take the derivative of the position function with respect to time. The derivative of a function $f(t)$ is denoted by $f'(t)$, and represents the rate of change of the function with respect to time.

The Derivative of the Position Function

To find the derivative of the position function $s(t) = \sqrt{2+2 \pi}$, we can use the chain rule. The chain rule states that if we have a composite function of the form $f(g(t))$, then the derivative of the composite function is given by $f'(g(t)) \cdot g'(t)$.

Applying the Chain Rule

In this case, we have $s(t) = \sqrt{2+2 \pi}$. We can rewrite this as $s(t) = (2+2 \pi)^{1/2}$. Now, we can apply the chain rule to find the derivative of $s(t)$.

The Derivative of $s(t)$

Using the chain rule, we have:

$$s'(t) = \frac{d}{dt} (2+2 \pi)^{1/2} = \frac{1}{2} (2+2 \pi)^{-1/2} \cdot \frac{d}{dt} (2+2 \pi)$$

Simplifying the Derivative

Now, we can simplify the derivative of $s(t)$.

$$s'(t) = \frac{1}{2} (2+2 \pi)^{-1/2} \cdot 0 = 0$$

The Velocity of the Particle

The derivative of the position function $s(t)$ is equal to the velocity of the particle. Therefore, the velocity of the particle is given by $s'(t) = 0$.

Conclusion

In this article, we have explored the position function of a particle moving along a coordinate line, and used it to find the particle's velocity at a given time. We have shown that the velocity of the particle is given by the derivative of the position function, and have found that the velocity of the particle is equal to zero.

Final Answer

The final answer is $\boxed{0}$.

Discussion

The position function $s(t) = \sqrt{2+2 \pi}$ describes the position of a particle at time $t$. The derivative of this function, $s'(t)$, represents the velocity of the particle. In this case, we have found that the velocity of the particle is equal to zero.

Why is the Velocity Zero?

The velocity of the particle is zero because the position function $s(t) = \sqrt{2+2 \pi}$ is a constant function. This means that the position of the particle does not change with time, and therefore the velocity of the particle is zero.

What is the Physical Meaning of Zero Velocity?

A velocity of zero means that the particle is not moving. In other words, the particle is at rest.

Conclusion

Q: What is the position function of a particle moving along a coordinate line?

A: The position function of a particle moving along a coordinate line is given by $s(t) = \sqrt{2+2 \pi}$, where $s$ is the position of the particle at time $t$.

Q: What is the meaning of the position function?

A: The position function $s(t) = \sqrt{2+2 \pi}$ describes the position of the particle at time $t$. It tells us that the position of the particle is constantly changing, and is dependent on the time $t$.

Q: How do we find the velocity of the particle?

A: To find the velocity of the particle, we need to take the derivative of the position function with respect to time. The derivative of a function $f(t)$ is denoted by $f'(t)$, and represents the rate of change of the function with respect to time.

Q: What is the derivative of the position function?

A: The derivative of the position function $s(t) = \sqrt{2+2 \pi}$ is given by $s'(t) = 0$. This means that the velocity of the particle is equal to zero.

Q: What does a velocity of zero mean?

A: A velocity of zero means that the particle is not moving. In other words, the particle is at rest.

Q: Why is the velocity of the particle zero?

A: The velocity of the particle is zero because the position function $s(t) = \sqrt{2+2 \pi}$ is a constant function. This means that the position of the particle does not change with time, and therefore the velocity of the particle is zero.

Q: What is the physical meaning of a constant position function?

A: A constant position function means that the particle is at rest. The position of the particle does not change with time, and therefore the particle is not moving.

Q: Can a particle have a constant position function and still be moving?

A: No, a particle cannot have a constant position function and still be moving. If the position function is constant, it means that the particle is at rest.

Q: What is the relationship between the position function and the velocity function?

A: The position function and the velocity function are related by the derivative. The derivative of the position function gives the velocity function.

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

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

Q: What is the significance of the position function in physics?

A: The position function is a fundamental concept in physics. It describes the position of an object at a given time, and is used to calculate the velocity and acceleration of the object.

Q: Can we use the position function to solve real-world problems?

A: Yes, we can use the position function to solve real-world problems. For example, we can use the position function to calculate the trajectory of a projectile, or to determine the position of an object in a given time.

Q: What are some common applications of the position function?

A: Some common applications of the position function include:

• Calculating the trajectory of a projectile
• Determining the position of an object in a given time
• Calculating the velocity and acceleration of an object
• Modeling the motion of a particle or a system of particles

Q: Can we use the position function to model complex systems?

A: Yes, we can use the position function to model complex systems. For example, we can use the position function to model the motion of a pendulum, or to determine the position of a satellite in orbit around the Earth.