Newton's Second Law — Question 4

Question 4

A force switch requires a piecewise motion history

Use $\sum\vec F=m\vec a$ for constant mass in an inertial frame. A 2 kg cart begins at rest at x=0. Net force is +4 N for 2 s, then -2 N for the following 3 s.

1. Find position and velocity at the switch.

2. Find the final position and velocity.

3. Does velocity jump when the finite force changes abruptly?

Question 4 - Solution

A force switch requires a piecewise motion history

(a) Solve the first stage.

$$a_1=4/2=2,\quad v_1=2(2)=4\,\mathrm{m/s},\quad x_1=\tfrac12(2)(2)^2=4\,\mathrm m$$ $a_1=2$ m/s$^2$, so at t=2 s, $v=4$ m/s and $x=4$ m.

(b) Use the switch state as new initial data.

$$a_2=-2/2=-1,\quad\boxed{v_f=4-3=1\,\mathrm{m/s}},\quad x_f=4+4(3)-\tfrac12(3)^2=11.5\,\mathrm m$$ $a_2=-1$ m/s$^2$. Over the next 3 s, $\Delta x=4(3)-\tfrac12(3)^2=7.5$ m and $v=1$ m/s. Final x is 11.5 m.

(c) Separate acceleration from velocity.

$$v(2^+)-v(2^-)=\lim_{\epsilon\to0}\frac1m\int_{2-\epsilon}^{2+\epsilon}F(t)\,dt=0$$ Acceleration changes instantly in this ideal piecewise model, but a bounded force acting for zero elapsed time produces no finite velocity jump. Position and velocity remain continuous at the switch.