Newton's First Law — Question 8

Question 8

A zero average change does not imply zero force throughout

A cart starts at 1 m/s east. It accelerates east at 2 m/s$^2$ for 1 s, then west at 2 m/s$^2$ for 1 s. Treat the track frame as inertial.

1. Find midpoint and final velocities.

2. Find the average acceleration for the full interval.

3. Does the zero average justify applying force-free coasting to the entire path?

Question 8 - Solution

A zero average change does not imply zero force throughout

(a) Track the two stages.

$$v(1)=1+2(1)=3,\quad\boxed{v(2)=3-2(1)=1\,\mathrm{m/s}}$$ Velocity rises to 3 m/s at 1 s and falls to 1 m/s at 2 s. The cart continues east in both stages.

(b) Use the endpoint definition.

$$a_{\mathrm{avg}}=\frac{v(2)-v(0)}{2}=\frac{1-1}{2}=0$$ $a_{\mathrm{avg}}=(1-1)/2=0$ m/s$^2$ over the full interval.

(c) Check what happened in between.

$$\Delta x=\frac{1+3}{2}(1)+\frac{3+1}{2}(1)=4\,\mathrm m\ne v_0(2)=2\,\mathrm m$$ Instantaneous acceleration is nonzero in both stages, so the net force is not continuously zero. Each stage covers 2 m, for total 4 m; coasting at the initial speed would cover only 2 m. Equal endpoint velocities do not establish uniform motion.