Reading a Potential Energy Curve — Question 9

Question 9

Inferring mass and energy from two curve readings

Assume one-dimensional classical motion with $K=E-U(x)$ and $F_x=-dU/dx$, with no dissipation. At one position a particle has U=2 J and speed 4 m/s; at another it has U=8 J and speed 2 m/s. Both observations belong to the same conservative motion.

1. Infer the mass and conserved total energy.

2. Find speed at a point with U=5 J, if that point lies in the accessible region.

3. Would an additional point with U=11 J be accessible at this energy?

Question 9 - Solution

Inferring mass and energy from two curve readings

(a) Equate the two total energies.

Equate the measured total energies: $$2+\tfrac12m(4)^2=8+\tfrac12m(2)^2\Rightarrow6m=6.$$ $$\boxed{m=1\ \mathrm{kg}},\qquad\boxed{E=2+8=10\ \mathrm J}.$$

(b) Use the newly inferred energy gap.

For an accessible point at $U=5$ J, $$K=10-5=5\ \mathrm J\Rightarrow\boxed{v=\sqrt{2(5)/1}=\sqrt{10}\ \mathrm{m/s}}.$$ The accessibility proviso excludes a disconnected region behind a higher barrier.

(c) Reject a negative kinetic energy.

At $U=11$ J, the required kinetic energy would be $$K=10-11=-1\ \mathrm J<0.$$ That point is classically inaccessible at this total energy.