Newtonian Mechanics — Question 3

Question 3

An accelerating camera can create apparent acceleration

A freely coasting bead has laboratory position $x=2+5t$ m. A camera origin follows $X=t^2$ m with parallel axes. Treat the laboratory as inertial and use numerical seconds.

1. Find the bead’s acceleration in each coordinate system.

2. Find when it appears momentarily at rest to the camera.

3. Explain why interpreting camera-coordinate acceleration as proof of a new bead interaction is invalid.

Question 3 - Solution

An accelerating camera can create apparent acceleration

(a) Subtract before differentiating.

$$x'=2+5t-t^2,\quad v'=5-2t,\quad \boxed{a'=0-2=-2\,\mathrm{m/s^2}}$$ $x^{\prime}=2+5t-t^2$ m, $v^{\prime}=5-2t$ m/s and $a^{\prime}=-2$ m/s$^2$. Laboratory acceleration is zero.

(b) Match velocities, not positions.

$$5-2t_*=0\ \Rightarrow\ t_*=2.5\,\mathrm s,\quad x'(t_*)=8.25\,\mathrm m$$ $v^{\prime}=0$ at 2.5 s. The bead is then at $x^{\prime}=8.25$ m, so apparent rest does not mean coincidence with the camera origin.

(c) Audit the frame.

$$F_{\mathrm{physical}}=0,\quad ma'=F_{\mathrm{physical}}-m\ddot X=-2m\,\mathrm{m/s^2}$$ The camera accelerates at $+2$ m/s$^2$. Its coordinate acceleration contributes $-2$ m/s$^2$ to the bead record even though the bead coasts in the laboratory. The simplest Newtonian force laws require an inertial frame; accelerating coordinates require additional treatment.