Newton's First Law — Question 7

Question 7

Equal-time samples support but do not prove uniform motion

At t=0, 1, 2 and 3 s a point has x=0, 2, 4 and 6 m. Two candidate histories are $x_A=2t$ m and $x_B=2t+0.1\sin(2\pi t)$ m, with numerical seconds.

1. Check both candidates against the samples.

2. Is either candidate accelerated between samples?

3. Can these samples alone prove zero net force at every time?

Question 7 - Solution

Equal-time samples support but do not prove uniform motion

(a) Evaluate integer times.

$$x_B(n)=2n+0.1\sin(2\pi n)=2n=x_A(n),\quad n=0,1,2,3$$ The sine term vanishes at each integer, so both histories produce all four reported positions.

(b) Differentiate the candidates.

$$v_B=2+0.2\pi\cos(2\pi t),\quad\boxed{a_B=-0.4\pi^2\sin(2\pi t)}$$ A has constant 2 m/s velocity. B has $v_B=2+0.2\pi\cos(2\pi t)$ m/s and $a_B=-0.4\pi^2\sin(2\pi t)$ m/s$^2$, generally nonzero between samples.

(c) Avoid an inference stronger than the data.

$$a_A=0,\qquad a_B(0.25)=-0.4\pi^2\,\mathrm{m/s^2}\ne0$$ The samples are consistent with first-law motion but do not uniquely establish it between measurements. Faster measurements or an independently justified model are needed to rule out unresolved variations.