A particle of mass 2 kg is subjected to a two dimensional conservative force Fx = 2x + 2y and Fy = 2x - y². If the particle has kinetic energy of (8/3) at point (2,3) find speed of particle when it reaches (1,2) [all values in SI units]

To find the speed of the particle when it reaches the point (1,2), we can use the principle of conservation of mechanical energy. In a conservative force field, the total mechanical energy (kinetic energy plus potential energy) remains constant. Let's break down the problem step by step.

Understanding the Forces

The forces acting on the particle are given by:

• Fx = 2x + 2y
• Fy = 2x - y²

These forces are conservative, meaning we can derive a potential energy function from them. The potential energy (U) can be found by integrating the force components.

Finding the Potential Energy

To find the potential energy function U(x, y), we can integrate the force components:

Integrating Fx

Starting with the x-component:

U(x, y) = -∫Fx dx = -∫(2x + 2y) dx = -[x² + 2xy] + g(y)

Here, g(y) is an arbitrary function of y that we will determine later.

Integrating Fy

Now, we integrate the y-component:

U(x, y) = -∫Fy dy = -∫(2x - y²) dy = -[2xy - (y³/3)] + h(x)

Here, h(x) is an arbitrary function of x.

Combining the Results

To find the complete potential energy function, we need to combine the results from both integrations. We can set:

U(x, y) = -x² - 2xy + (y³/3) + C

where C is a constant that can be determined later if needed.

Calculating the Total Mechanical Energy

The total mechanical energy (E) at any point is given by:

E = K + U

Where K is the kinetic energy. At the point (2,3), the kinetic energy is given as:

K = 8/3 J

Now, we need to calculate the potential energy at this point:

U(2, 3) = -2² - 2(2)(3) + (3³/3) = -4 - 12 + 9 = -7 J

Now we can find the total mechanical energy at (2,3):

E = K + U = (8/3) + (-7) = (8/3) - (21/3) = -13/3 J

Finding the Speed at (1,2)

Next, we need to find the potential energy at the point (1,2):

U(1, 2) = -1² - 2(1)(2) + (2³/3) = -1 - 4 + (8/3) = -5 + (8/3) = -15/3 + (8/3) = -7/3 J

Now we can calculate the total mechanical energy at (1,2):

E = K + U

Substituting the known values:

-13/3 = K + (-7/3)

K = -13/3 + 7/3 = -6/3 = -2 J

Now, we can find the speed (v) using the kinetic energy formula:

K = (1/2)mv²

-2 = (1/2)(2)v²

-2 = v²

Since kinetic energy cannot be negative, we must have made an error in our calculations. Let's re-evaluate the potential energy calculations or the total energy conservation. However, if we assume the calculations are correct, we can conclude that the speed at point (1,2) cannot be determined from the given information, as the kinetic energy cannot be negative.

In summary, the speed of the particle cannot be calculated as it leads to an inconsistency in the energy values. Please check the values or the calculations for any discrepancies.