Conservative force, kinetic energy

[nothingatall]

Homework Statement

A single conservative force F(x) acts on a 2.4 kg particle that moves along an x axis. The potential energy U(x) associated with F(x) is given by
U(x) = -4xe-x/4
where x is in meters. At x = 5.0 m the particle has a kinetic energy of 5.2 J. (a) What is the mechanical energy of the system? (b) What is the maximum kinetic energy of the particle and (c) the value of x at which it occurs?

Homework Equations

Emech=K+U

The Attempt at a Solution

I know how to find the mechanical energy, but I don't know how to do the derivative of U(x) so that I can do b) and c) even though its simple.

 

[willem2]

this should be enough to find it

the derivative of e^x is e^x
the derivative of f(x)g(x) = f'(x)g(x) + f(x)g'(x)
the derivative of f(g(x)) = g'(x) f'(g(x))

 

[tomcue]

As a scientist, it is important to have a strong understanding of the fundamental equations and principles in physics. In this scenario, we are dealing with a conservative force and kinetic energy. A conservative force is one that does work on an object that is independent of the path taken by the object. This means that the work done by the force only depends on the initial and final positions of the object, and not on the path it took to get there.

In this problem, we are given the potential energy function U(x) and we are asked to find the mechanical energy of the system, as well as the maximum kinetic energy and the value of x at which it occurs. To do this, we can use the equation for mechanical energy, Emech = K + U, where K is the kinetic energy and U is the potential energy.

To find the mechanical energy, we need to find the kinetic energy at x = 5.0 m. We are given that the kinetic energy at this point is 5.2 J, so we can plug this into the equation:

Emech = K + U = 5.2 J + (-4 * 5.0 m * e^(-5.0 m/4)) = 5.2 J - 5.2 J = 0 J

This means that the mechanical energy of the system is 0 J. This makes sense, as the potential energy is always equal to the negative of the kinetic energy for a conservative force.

To find the maximum kinetic energy, we can use the fact that the maximum kinetic energy occurs when the potential energy is at its minimum value. This can be found by taking the derivative of the potential energy function and setting it equal to 0:

dU/dx = -4e^(-x/4) + (4/4)xe^(-x/4) = 0

Solving for x, we get x = 4 m. This means that the maximum kinetic energy occurs at x = 4 m. To find the value of the maximum kinetic energy, we can plug this value of x into the potential energy function:

U(x=4 m) = -4 * 4 m * e^(-4 m/4) = -4 J

Therefore, the maximum kinetic energy is 4 J.

In summary, the mechanical energy of the system is 0 J, the maximum kinetic energy is 4 J, and it occurs at