Motion — Question 2

Question 2

A point model is a choice made for a task

A rigid bus is $12.0\ \mathrm{m}$ long. It travels forward on a straight road without turning. A tracker reports the midpoint between its front and rear exactly. For a point model, the whole bus is represented by that midpoint.

Consider two tasks:

• A route map may locate the bus by a representative point if ignoring its extent puts every part of the bus within $10.0\ \mathrm{m}$ of the reported point.

• A tunnel monitor must determine whether the entire bus has passed a fixed exit plane, with a location tolerance of $0.20\ \mathrm{m}$.

All dimensions are exact within this model. Ignore tracking errors and the bus’s width and height.

1. Find the greatest distance from the tracked midpoint to any part of the bus along the road. Does the proposed point representation satisfy the route-map criterion?

2. At the instant the midpoint reaches the tunnel exit, where are the front and rear relative to that plane? Has the entire bus cleared the tunnel?

3. How much farther must the midpoint travel before the rear reaches the exit? Compare the omitted length with the tunnel monitor’s tolerance.

4. Is it necessary to abandon the midpoint tracker for the tunnel task? Describe a sufficient model improvement. Explain why “large objects cannot be particles” is not a useful universal rule for modeling motion.

Question 2 - Solution

A point model is a choice made for a task

(a) Compare the neglected extent with the task’s criterion.

Let $L=12.0\ \mathrm{m}$ and measure $x_c,x_f,x_r$ for the centre, front and rear from the exit plane, positive forward. $$\max|x-x_c|=\frac L2=6\ \mathrm{m}<10\ \mathrm{m}.$$

The midpoint is half a bus length from either end, so the greatest longitudinal separation is $12.0/2=\boxed{6.0\ \mathrm{m}}$. Since $6.0\ \mathrm{m}<10.0\ \mathrm{m}$, the point representation satisfies the stated route-map criterion. This does not claim that the bus is physically point-sized.

(b) A midpoint event is not a rear-end event.

$$x_f=x_c+6\ \mathrm{m},\quad x_r=x_c-6\ \mathrm{m};\qquad x_c=0\Rightarrow(x_f,x_r)=(6,-6)\ \mathrm{m}.$$

When the midpoint lies in the exit plane, the front is 6.0 m beyond it and the rear is 6.0 m short of it. The bus therefore still occupies part of the tunnel. Replacing the bus by a point would identify the wrong event if midpoint passage were labeled complete clearance.

(c) Retain the dimension controlling clearance.

$$x_r\geq0\iff x_c\geq6\ \mathrm{m};\qquad\frac{6\ \mathrm{m}}{0.20\ \mathrm{m}}=30.$$

The midpoint must move another $\boxed{6.0\ \mathrm{m}}$ for the rear to reach the exit. The entire bus is strictly beyond the exit just after that event. The neglected 6.0 m is $6.0/0.20=\boxed{30}$ times the location tolerance, so it cannot be ignored for this task.

(d) Improve the model only as much as the task requires.

The midpoint tracker can be retained. Add the known half-length and orientation: the rear remains 6.0 m behind the midpoint. Monitor rear passage using that relation. Full details of seats, wheels, or engine motion are unnecessary for this clearance question.

Whether an object may be treated as a particle depends on the quantity and accuracy sought. The same bus can be adequately represented by one point on the route map while requiring a finite-length model at the exit. A point model describes selected motion; it does not assert that the object’s size vanishes.