Finding a constant in a particle motion problem using the energy equation

Question

I have found V in terms of x, and then I have found the energy equation for x=1 and x=2. I've then set them equal to one another and solved, finding lambda = 20. I didn't use the values v=4 and v=2; since they are provided, I assume my method is incorrect/incomplete?

Answer 1

Potential energy, alone, is not constant. It is the sum of potential and kinetic energy, which is a constant along the motion.

Answer 2

Only kinetic+potential energy is constant. You should not expect the kinetic energy to be constant. Instead, the change in the kinetic energy is $\int_1^2 F dx$, and it is also $\frac{1}{2} m \left ( v_2^2 - v_1^2 \right )$. Can you solve the resulting equation for $\lambda$?