Work and Kinetic Energy — Question 3

Question 3

A favorable endpoint cannot repair an inaccessible intermediate state

Use the work–kinetic-energy theorem $W_{\mathrm{net}}=\Delta K$ for the specified particle in an inertial frame. A particle starts with kinetic energy 5 J at x=0 and is proposed to move continuously to x=2 m. Net cumulative work from the start is piecewise linear through the points $(x,W)=(0,0),(1,-8),(2,2)$, in metres and joules.

1. Compute the formal endpoint kinetic energy at x=2 m.

2. Find the first x where the kinetic energy reaches zero.

3. Explain why the endpoint result does not prove that x=2 m can be reached from the stated start.

Question 3 - Solution

A favorable endpoint cannot repair an inaccessible intermediate state

(a) Use the endpoint ledger.

The governing relation is $$\boxed{K(x)=K_i+W(x)\geq0\quad\text{at every accessible point}}$$ The formal result is $$K(2)=5+2=7\ \mathrm J.$$

(b) Check the first segment.

For $$0\le x\le1,$$

$$W=-8x\ \mathrm J.$$ Thus $K=5-8x$ reaches zero at $$x=5/8=0.625\ \mathrm m.$$

(c) Enforce feasibility along a continuous route.

The proposed forward path would require negative K immediately beyond 0.625 m. With the stated work profile and no additional interaction the particle cannot traverse that interval. Later positive work is unavailable until the particle reaches the region where it would occur.