Kinetic Energy of a Particle with Given Force

The Given Information:
A particle of mass M is subjected to a force F = F(cos(theta)i + sin(theta)j), where F is the magnitude of the force and theta is the angle between the force and the positive x-axis.

Finding the Kinetic Energy:
To find the kinetic energy of the particle as a function of time, we need to use Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration.

Step 1: Finding the Acceleration:
From Newton's second law, we know that the net force acting on the particle is equal to its mass multiplied by its acceleration. In this case, the net force acting on the particle is the given force F. Therefore, we can equate the given force to the mass of the particle (M) multiplied by its acceleration (a).

F = Ma

Step 2: Resolving the Force:
The given force can be resolved into its x and y components using the trigonometric identities for cosine and sine functions. The x-component of the force is Fcos(theta) and the y-component is Fsin(theta).

Step 3: Equating the Forces:
Since the acceleration is a vector quantity, we need to equate the x and y components of the given force to the respective components of the mass times acceleration. This gives us two equations:

Fcos(theta) = Ma_x
Fsin(theta) = Ma_y

Step 4: Solving for Acceleration:
From the above equations, we can solve for the x and y components of the acceleration. Dividing both sides of the equations by mass gives:

a_x = Fcos(theta)/M
a_y = Fsin(theta)/M

Step 5: Finding the Kinetic Energy:
The kinetic energy (KE) of a particle is given by the equation KE = (1/2)MV^2, where V is the magnitude of the velocity of the particle. To find the velocity, we need to integrate the acceleration with respect to time.

Integrating Acceleration:
Since acceleration is the rate of change of velocity, integrating the x and y components of acceleration with respect to time gives the x and y components of the velocity.

v_x = ∫ a_x dt
v_y = ∫ a_y dt

Step 6: Finding the Magnitude of Velocity:
The magnitude of the velocity is given by the equation V = sqrt(v_x^2 + v_y^2). Substituting the expressions for v_x and v_y from step 5 into this equation gives:

V = sqrt((∫ a_x dt)^2 + (∫ a_y dt)^2)

Step 7: Calculating the Kinetic Energy:
Finally, substituting the expression for V into the equation for kinetic energy, we obtain the kinetic energy as a function of time:

KE = (1/2)M(sqrt((∫ a_x dt)^2 + (∫ a_y dt)^2))^2

This is the equation for the kinetic energy of