Reading a Potential Energy Curve — Question 1

Question 1

Reading slopes and kinetic-energy gaps

Assume one-dimensional classical motion with $K=E-U(x)$ and $F_x=-dU/dx$, with no dissipation. A potential graph joins (0 m,6 J), (2 m,2 J), and (4 m,6 J) by straight lines. A 2 kg particle has E=8 J.

1. Find force at x=1 and x=3 m.

2. Find speed at x=1 m and the largest speed on the plotted interval.

3. Explain why the point of largest speed need not have a well-defined derivative in this idealized graph.

Question 1 - Solution

Reading slopes and kinetic-energy gaps

(a) Use the negative segment slopes.

Take the negative slope on each straight segment: $$\boxed{F(1)=-\frac{2-6}{2-0}=+2\ \mathrm N},\qquad\boxed{F(3)=-\frac{6-2}{4-2}=-2\ \mathrm N}.$$

(b) Read the vertical energy gap.

On the left segment, $U(1)=4$ J: $$\boxed{v(1)=\sqrt{2(8-4)/2}=2\ \mathrm{m/s}}.$$ At the minimum $U(2)=2$ J, $$\boxed{v_{\max}=\sqrt{2(8-2)/2}=\sqrt6\ \mathrm{m/s}}.$$

(c) Separate energy continuity from force smoothness.

The continuous energy gap fixes a continuous speed: $$v(x)=\sqrt{2[E-U(x)]/m}.$$ A corner in U changes its one-sided slopes and therefore the force, but does not create a jump in U or speed. The speed maximum exists even without a unique derivative at the corner.