Newton's First Law — Question 4

Question 4

An instant of rest is not a first-law state

A bead has $x(t)=(t-2)^2$ m with numerical seconds near t=2 s, in an inertial frame.

1. Find velocity and acceleration at 2 s.

2. Can the bead’s net force be zero throughout a neighborhood of that instant?

3. Compare this event with a bead that remains at x=0.

Question 4 - Solution

An instant of rest is not a first-law state

(a) Differentiate the history.

$$v=2(t-2),\quad \boxed{v(2)=0,\quad a(2)=2\,\mathrm{m/s^2}}$$ $v=2(t-2)$ m/s and $a=2$ m/s$^2$. At 2 s the velocity is zero while acceleration is nonzero.

(b) Use an interval, not a snapshot.

$$F_{{\mathrm{net}},x}=ma=2m\,\mathrm{m/s^2}\ne0$$ It reverses direction through that instant. Constant velocity does not hold in any surrounding interval, so the history is not a zero-net-force first-law motion.

(c) Distinguish lasting rest.

$$x_{\mathrm{fixed}}(t)=0\ \Rightarrow\ v_{\mathrm{fixed}}=0,\quad a_{\mathrm{fixed}}=0$$ A bead staying at x=0 has velocity and acceleration zero throughout the interval. A single photograph of either at x=0 could look the same; neighboring observations distinguish their dynamical states.