Analyzing the Velocity and Acceleration of a Moving Particle

In the field of physics, understanding the motion of particles is crucial. This article will walk you through the process of finding the velocity and acceleration of a particle given its coordinates as functions of time. Specifically, we will analyze a particle with the coordinates given by x 4t - t^2 / 2 and y 36t - t^3 / 6 at t 2.

Understanding the Coordinates and their Derivatives

The coordinates of a moving particle are given by:

x 4t - t^2 / 2 y 36t - t^3 / 6

To find the velocity and acceleration of the particle, we need to take the first and second derivatives of these position functions with respect to time t.

Calculating the Velocity

The velocity of the particle is given by the first derivatives of the position functions:

The velocity along the x-component: v_x dx/dt 4 - t The velocity along the y-component: v_y dy/dt 6 - t^2 / 2

At t 2:

v_x(2) 4 - 2 2 v_y(2) 6 - (2^2) / 2 6 - 2 4

The velocity of the particle at t 2 is thus:

v_{t2} 2i 4j

Calculating the Acceleration

The acceleration of the particle is given by the second derivatives of the velocity functions:

The acceleration along the x-component: a_x dv_x/dt -1 The acceleration along the y-component: a_y dv_y/dt -t

At t 2:

a_x(2) -1 a_y(2) -2

The acceleration of the particle at t 2 is thus:

a_{t2} -i - 2j

Putting It All Together

By following the steps above, we can summarize the findings:

The velocity of the particle at t 2 is 2^ 25 The acceleration of the particle at t 2 is 5^

Understanding these calculations is essential for further analysis of particle motion and can be applied in various fields such as engineering and physics.