Newtonian Mechanics — Question 1

Question 1

A motion record does not identify its cause

A cart has $x(t)=2+3t+t^2$ m, with numerical t in seconds, for $0\le t\le3$ s. One account attributes its acceleration to a tow rope; another to a motor acting through a track. No interaction measurements are supplied.

1. Find initial position, velocity and acceleration.

2. Do the position data alone distinguish the two accounts?

3. Propose an additional observation that would discriminate between the accounts.

Question 1 - Solution

A motion record does not identify its cause

(a) Extract only what the record fixes.

$$x_0=x(0)=2\,\mathrm m,\quad v=\frac{dx}{dt}=3+2t,\quad \boxed{a=\frac{d^2x}{dt^2}=2\,\mathrm{m/s^2}}$$ $x_0=2$ m, $v=(3+2t)$ m/s and $a=2$ m/s$^2$. At 3 s, $x=20$ m and $v=9$ m/s.

(b) Separate description from explanation.

$$F_{\mathrm{net}}=ma=2m\,\mathrm{m/s^2},\qquad F_{\mathrm{net}}=F_{\mathrm{rope}}+F_{\mathrm{track}}+\cdots$$ The same trajectory can result from different arrangements of interactions. Its acceleration is fixed, but the record does not label which agent produced it. A dynamical model needs an identified body, its interactions and an appropriate reference frame.

(c) Test the proposed mechanism.

$$\Delta F_{\mathrm{net}}=m\Delta a\quad(m\text{ fixed})$$ Inspect or measure the rope tension while isolating the motor, or repeat with the rope disconnected while holding the motor conditions fixed. The intervention must change the proposed cause in a controlled way; another sample of the same unperturbed trajectory cannot by itself identify that cause.