Newtonian Mechanics — Question 5

Question 5

A particle model has a question-dependent domain

A cart is 0.40 m long. Its center is 10.0 m from a finish line and moves at a steady 2.0 m/s toward it. The cart does not rotate; the finish line is perpendicular to the motion.

1. Find when the center crosses the line.

2. Find when the leading edge crosses it.

3. Explain when treating the cart as a point is sufficient and when it discards required information.

Question 5 - Solution

A particle model has a question-dependent domain

(a) Track the chosen representative point.

$$x_C(t)=vt,\quad vt_C=d\ \Rightarrow\ \boxed{t_C=\frac{10.0}{2.0}=5.00\,\mathrm s}$$ Center crossing takes $10/2=5.00$ s. A point model placed at the center directly answers that question.

(b) Restore the finite geometry.

$$vt_F+\frac L2=d\ \Rightarrow\ \boxed{t_F=\frac{10.0-0.20}{2.0}=4.90\,\mathrm s}$$ The leading edge begins 0.20 m closer, so its travel is 9.80 m and its crossing time is 4.90 s. The difference is 0.10 s.

(c) Tie simplification to the observable.

$$\Delta t=t_C-t_F=\frac{L}{2v}=0.10\,\mathrm s$$ A point model can describe center translation while omitting shape. Edge-clearance, rotation or contact-location questions require geometric information beyond that one position. Small size alone does not justify ignoring a dimension when that dimension controls the event being timed.