Work Done by a General Variable Force — Question 7

Question 7

A finite force step does not create a kinetic-energy jump

A 1 kg particle starts at x=0 moving right at 2 m/s. The net force is $+4$ N for $0\le x<1$ m and $-2$ N for $1<x\le3$ m. The value at x=1 itself is finite but otherwise arbitrary.

1. Find kinetic energy just before and just after x=1 m.

2. Find speed at x=3 m.

3. Explain which of force, acceleration, velocity, and kinetic energy can jump at the boundary.

Question 7 - Solution

A finite force step does not create a kinetic-energy jump

(a) A single point contributes no finite area.

The governing relation is $$\boxed{K(x)=K_0+\int_0^xF_x(s)\,ds}$$ Work through x=1 is 4 J, so $$K(1^-)=K(1^+)=2+4=6\ \mathrm J.$$ Changing a bounded force at one point does not change its integral.

(b) Subtract work over the second interval.

$$K(3)=6-2(2)=2\ \mathrm J,$$ so $$v(3)=2\ \mathrm{m/s}.$$ K remains positive on the route.

(c) Distinguish a force step from an impulse.

Force and acceleration jump from +4 to $-2$ in their respective units. Velocity and kinetic energy remain continuous. A finite step is not an infinite impulse acting in zero time.