JEE PYQ: Motion in a Straight Line - Question ID d7df2d9d9e1c (JEE Main 2019)

When two objects of finite length move along the same straight line, the time taken for one to completely cross the other depends on their relative speed and the total distance that must be covered.

• Relative speed:
  • Same direction: $v_{\text{rel}} = |v_1 - v_2|$
  • Opposite direction: $v_{\text{rel}} = v_1 + v_2$
• Total distance to be covered for complete crossing is the sum of the lengths of the two trains: $L_{\text{total}} = L_1 + L_2$
• Time taken to cross: $t = \frac{L_{\text{total}}}{v_{\text{rel}}}$

Given data:
Passenger train (P): length $L_P = 60$ m, speed $v_P = 80$ km/h
Freight train (F): length $L_F = 120$ m, speed $v_F = 30$ km/h

Step 1 – Convert speeds to m/s
$v_P = 80 \times \frac{1000}{3600} = \frac{800}{36} = \frac{200}{9}$ m/s
$v_F = 30 \times \frac{1000}{3600} = \frac{300}{36} = \frac{25}{3}$ m/s

Step 2 – Compute relative speeds
(i) Same direction:
$v_{\text{rel,same}} = v_P - v_F = \frac{200}{9} - \frac{25}{3} = \frac{200 - 75}{9} = \frac{125}{9}$ m/s
(ii) Opposite direction:
$v_{\text{rel,opp}} = v_P + v_F = \frac{200}{9} + \frac{25}{3} = \frac{200 + 75}{9} = \frac{275}{9}$ m/s

Step 3 – Total distance to cross
$L_{\text{total}} = L_P + L_F = 60 + 120 = 180$ m

Step 4 – Compute crossing times
(i) Same direction:
$t_{\text{same}} = \frac{180}{\frac{125}{9}} = \frac{180 \times 9}{125} = \frac{1620}{125} = \frac{324}{25}$ s
(ii) Opposite direction:
$t_{\text{opp}} = \frac{180}{\frac{275}{9}} = \frac{180 \times 9}{275} = \frac{1620}{275} = \frac{324}{55}$ s

Step 5 – Form the ratio
$\frac{t_{\text{same}}}{t_{\text{opp}}} = \frac{\frac{324}{25}}{\frac{324}{55}} = \frac{55}{25} = \frac{11}{5}$

Hence the ratio of times is $\frac{11}{5}$, which corresponds to option C.

1. Unit mismatch: Forgetting to convert km/h to m/s leads to incorrect relative speeds. 2. Relative speed sign: In same-direction motion, students sometimes add speeds instead of subtracting. 3. Total distance: Some mistakenly use only one train’s length instead of the sum $L_P + L_F$. 4. Ratio order: The question asks for $t_{\text{same}} : t_{\text{opp}}$; reversing the ratio gives $\frac{5}{11}$, which is not among the options.

Tip: Always draw a quick sketch of the two trains and mark the relative motion arrows to avoid confusion.