Understanding the Motion of a Particle and Calculating Average Velocity

In this article, we will delve into a specific example of a particle's motion along the x-axis, deriving the position function, velocity function, and ultimately calculating the average velocity during a specified period. This example involves the application of calculus principles and is aimed at students and professionals in physics and engineering who need to understand particle motion and its mathematical representation.

Position Function of the Particle

The position of a particle moving along the x-axis is given by the function x 2122t - 6t^2, where t is the time in seconds. This function describes how the position of the particle changes with time. To better understand this, let's break down the components of the position function:

The term 2122t represents a linear component, which means the position increases at a rate proportional to the time. On the other hand, the term -6t^2 is a quadratic term, which introduces a deceleration or acceleration component. These components combine to give a dynamic motion to the particle.

Determining the Velocity Function

To find the velocity of the particle at any given time, we need to take the derivative of the position function with respect to time. This is done as follows:

dx/dt velocity 22 - 12t m/s

The velocity function, v(t) 22 - 12t, tells us how the velocity of the particle changes with respect to time. At different times, the velocity can increase or decrease, reflecting the changing nature of the particle's motion.

Evaluating Velocity at Specific Points

We can now calculate the velocity of the particle at specific points in time. Let's consider t 1 second and t 3 seconds:

At t 1 second: v(1) 22 - 12(1) 10 m/s At t 3 seconds: v(3) 22 - 12(3) -14 m/s

The velocity at t 3 seconds is negative, indicating that the particle is moving in the opposite direction. This change in direction is an important feature of particle motion and is often associated with the turning point of a particle's trajectory.

Calculating the Average Velocity

The average velocity over a period can be calculated by taking the total displacement and dividing it by the total time. In this case, the average velocity over the interval from t 1 second to t 3 seconds is:

Average velocity (velocity at 1 second velocity at 3 seconds) / 2

Average velocity (10 (-14)) / 2 -2 m/s

The negative value of the average velocity indicates that the particle is moving towards the negative x-axis, reflecting the change in direction during the interval.

Conclusion

This article has provided a clear understanding of how to determine the velocity of a particle given its position function, and how to calculate the average velocity over a specific time interval. By applying the principles of calculus, we can analyze and predict the motion of particles with precision. This knowledge is crucial in various scientific and engineering applications, making it a valuable tool for students and professionals alike.