A single conservative force F(x) acts on a 1.6 kg particle that moves along an x axis. The potential energy U(x) associated with F(x) is given by
U(x) = -2.9xe-x/3
where x is in meters. At x = 3.4 m the particle has a kinetic energy of 7.2 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?
To find the mechanical energy of the system, we need to add the kinetic energy (K) and the potential energy (U) together.
(a) The mechanical energy (E) of the system is given by:
E = K + U
We know the kinetic energy, which is given as 7.2 J. Let's calculate the potential energy at x = 3.4 m using the given equation:
U(x) = -2.9xe^(-x/3)
Substituting x = 3.4 m into the equation:
U(3.4) = -2.9(3.4)e^(-3.4/3)
Now, we have both the kinetic energy and the potential energy at x = 3.4 m. We can calculate the mechanical energy of the system:
E = K + U(3.4)
(b) To find the maximum kinetic energy of the particle, we need to determine the point where the kinetic energy is at its maximum. This occurs when the potential energy is at its minimum, which corresponds to the point where the particle is in stable equilibrium. At this point, all the potential energy is converted to kinetic energy.
(c) To find the value of x at which the maximum kinetic energy occurs, we need to find the minimum point of the potential energy function. We can do this by finding the value of x where the derivative of the potential energy function with respect to x is zero.
dU(x)/dx = 0
Taking the derivative of the potential energy equation:
dU(x)/dx = -2.9e^(-x/3) + 0.96xe^(-x/3)
Setting the derivative equal to zero and solving for x will give us the value of x at which the maximum kinetic energy occurs.