Newtonian Mechanics — Question 9

Question 9

Constrain a model with multiple observations

At t=0, 1 and 2 s a cart is at x=0, 2 and 6 m. A model assumes constant acceleration. An independent observation at 3 s gives x=11 m, with all data exact for this exercise.

1. Infer initial velocity and acceleration from the first three observations.

2. Predict x at 3 s and compare.

3. What does the disagreement establish, and what does it not establish?

Question 9 - Solution

Constrain a model with multiple observations

(a) Solve the two increments.

$$v_0+\tfrac12a=2,\quad2v_0+2a=6\ \Rightarrow\ \boxed{a=2,\ v_0=1}\quad\text{(SI)}$$ $v_0+\tfrac12a=2$ and $2v_0+2a=6$ in SI numerical values. Thus $a=2$ m/s$^2$ and $v_0=1$ m/s.

(b) Use a withheld observation.

$$x_{\mathrm{fit}}(3)=3+9=12\,\mathrm m,\quad r=11-12=-1\,\mathrm m$$ The model predicts $x(3)=3+9=12$ m, one metre larger than the reported 11 m. It cannot fit all four exact records with one constant acceleration.

(c) Reject only the tested claim.

$$\Delta^2x_0=6-2(2)+0=2,\quad\Delta^2x_1=11-2(6)+2=1\quad\mathrm m$$ This falsifies that constant-acceleration fit under the stated exact-data assumptions. It does not by itself refute Newtonian mechanics, which permits changing interactions and changing acceleration. In a real experiment measurement errors and frame assumptions would also require examination.