Relative Motion in One Dimension — Question 4

Question 4

Passing an extended object requires clearing both lengths

Train A is 120 m long and moves east at 25 m/s. Train B is 180 m long and moves east at 15 m/s on a parallel track. Timing begins when A’s front is level with B’s rear.

1. Find the time until A’s rear clears B’s front.

2. Check the result using ground positions with both initial touching endpoints at x=0.

3. If B instead travels west at 15 m/s and their fronts first align at the start, find the time for complete passing.

Question 4 - Solution

Passing an extended object requires clearing both lengths

(a) Measure the required relative advance.

$$(v_A-v_B)t=L_A+L_B\ \Rightarrow\ \boxed{t=\frac{120+180}{25-15}=30\,\mathrm s}$$ A must advance relative to B by $120+180=300$ m. Relative speed is $25-15=10$ m/s, so complete passing takes 30 s. Using only one train length would stop the timing before both trains clear.

(b) Check both critical endpoints.

$$x_{A,\mathrm{rear}}=25(30)-120=630\,\mathrm m=x_{B,\mathrm{front}}=180+15(30)$$ At 30 s, A’s front is at 750 m and its rear at $750-120=630$ m. B’s front is at $180+15(30)=630$ m. These endpoints are exactly level at the completion threshold.

(c) Add opposing speeds with their signs.

$$t_{\mathrm{opposite}}=\frac{L_A+L_B}{v_A-(-|v_B|)}=\frac{300}{40}=7.5\,\mathrm s$$ For opposite motion, relative velocity is $25-(-15)=40$ m/s. From front alignment to rear clearance the required relative travel is still 300 m, giving $300/40=7.5$ s. Train lengths are spatial extents; their role does not disappear when the directions change.