Trains Crossing Each Other
When two trains cross each other, the key idea is relative speed. The total distance to be covered is the sum of the lengths of both trains, and the time taken depends on whether the trains move in opposite directions (speeds add) or in the same direction (speeds subtract).
Core formulas
- Opposite directions: relative speed = ##v_1 + v_2##
- Same direction: relative speed = ##v_1 – v_2##
- Time to cross = ##\dfrac{L_1 + L_2}{\text{relative speed}}##
This quiz tests your ability to apply these ideas to medium-level competitive exam problems. Convert units carefully: ##1\ \text{km/h} = \dfrac{5}{18}\ \text{m/s}##.
Q1. Two trains of lengths 120 m and 180 m move in opposite directions at 54 km/h and 36 km/h. What is the total distance to be covered for them to cross each other?
When two trains cross each other, the total distance covered is the sum of their lengths: ##120 + 180 = 300## m.
Q2. Two trains move in opposite directions at 40 km/h and 50 km/h. What is their relative speed?
For opposite directions, relative speed is the sum: ##40 + 50 = 90## km/h.
Q3. Two trains move in the same direction at 72 km/h and 54 km/h. What is their relative speed?
For same direction, relative speed is the difference: ##72 - 54 = 18## km/h.
Q4. Two trains of lengths 150 m and 250 m cross each other in opposite directions at 60 km/h and 48 km/h. Find the time taken to cross.
Relative ##speed = 60 + 48 = 108## ##km / h = 108 \times \dfrac{5}{18} = 30## m/s. Total ##distance = 150 + 250 = 400## m. ##Time = \dfrac{400}{30} = \dfrac{40}{3}## s.
Q5. Two trains of equal length 200 m each move in the same direction at 90 km/h and 54 km/h. How long does the faster train take to cross the slower one?
Relative ##speed = 90 - 54 = 36## ##km / h = 36 \times \dfrac{5}{18} = 10## m/s. Total ##distance = 200 + 200 = 400## m. ##Time = \dfrac{400}{10} = 40## s.
Q6. Two trains of lengths 100 m and 150 m cross each other in 10 s while moving in opposite directions. If the first train moves at 36 km/h, find the speed of the second train.
Total ##distance = 100 + 150 = 250## m. Relative ##speed = \dfrac{250}{10} = 25## ##m / s = 25 \times \dfrac{18}{5} = 90## km/h. So ##36 + v_2 = 90##, giving ##v_2 = 54## km/h.
Q7. A 200 m train moving at 72 km/h crosses another train of length 300 m moving in the opposite direction at 36 km/h. Find the time taken.
Relative ##speed = 72 + 36 = 108## ##km / h = 30## m/s. Total ##distance = 200 + 300 = 500## m. ##Time = \dfrac{500}{30} = \dfrac{50}{3}## s.
Q8. Two trains of lengths 180 m and 220 m move in the same direction at 60 km/h and 42 km/h. Find the time taken by the faster train to cross the slower one.
Relative ##speed = 60 - 42 = 18## ##km / h = 5## m/s. Total ##distance = 180 + 220 = 400## m. ##Time = \dfrac{400}{5} = 80## s.
Q9. When two trains cross each other, the total distance to be covered equals:
To completely cross each other, the combined length of both trains must pass a given point, so the total distance is the sum of the two lengths.
Q10. Two trains of lengths 240 m and 160 m cross each other in 20 s while moving in opposite directions. What is their relative speed?
Total ##distance = 240 + 160 = 400## m. Relative ##speed = \dfrac{400}{20} = 20## m/s.
Q11. Two trains of lengths 150 m and 250 m move in the same direction at 80 km/h and 44 km/h. Find the time taken by the faster train to cross the slower one.
Relative ##speed = 80 - 44 = 36## ##km / h = 10## m/s. Total ##distance = 150 + 250 = 400## m. ##Time = \dfrac{400}{10} = 40## s.
Q12. Two trains of equal length cross each other in 12 s while moving in opposite directions at 54 km/h and 36 km/h. Find the length of each train.
Relative ##speed = 54 + 36 = 90## ##km / h = 25## m/s. Total ##distance = 25 \times 12 = 300## m. Since lengths are equal, each train is ##\dfrac{300}{2} = 150## m.
Q13. A train of length 300 m moving at 90 km/h crosses another train of length 200 m moving in the same direction at 54 km/h. Find the time taken.
Relative ##speed = 90 - 54 = 36## ##km / h = 10## m/s. Total ##distance = 300 + 200 = 500## m. ##Time = \dfrac{500}{10} = 50## s.
Q14. Two trains of lengths 120 m and 180 m cross each other in 15 s while moving in opposite directions. If one train moves at 48 km/h, find the speed of the other train.
Total ##distance = 120 + 180 = 300## m. Relative ##speed = \dfrac{300}{15} = 20## ##m / s = 72## km/h. So ##48 + v_2 = 72##, giving ##v_2 = 24## km/h.
Q15. Two trains of lengths 200 m and 300 m move in the same direction at 72 km/h and 36 km/h. How long does the faster train take to cross the slower one?
Relative ##speed = 72 - 36 = 36## ##km / h = 10## m/s. Total ##distance = 200 + 300 = 500## m. ##Time = \dfrac{500}{10} = 50## s.
Q16. Two trains of lengths 160 m and 240 m cross each other in 16 s while moving in opposite directions. What is their relative speed in km/h?
Total ##distance = 160 + 240 = 400## m. Relative ##speed = \dfrac{400}{16} = 25## ##m / s = 25 \times \dfrac{18}{5} = 90## km/h.
Q17. Two trains of lengths 250 m and 350 m move in the same direction at 66 km/h and 30 km/h. Find the time taken by the faster train to cross the slower one.
Relative ##speed = 66 - 30 = 36## ##km / h = 10## m/s. Total ##distance = 250 + 350 = 600## m. ##Time = \dfrac{600}{10} = 60## s.
Q18. Two trains of lengths 180 m and 220 m cross each other in 20 s while moving in opposite directions. If their speeds are in the ratio ##2:3##, find the speed of the faster train.
Total ##distance = 180 + 220 = 400## m. Relative ##speed = \dfrac{400}{20} = 20## m/s ##= 72## km/h. Since the speeds are in the ratio ##2:3##, the faster train has speed ##\dfrac{3}{5} \times 72 = 43.2## km/h.
Q19. Two trains of lengths 140 m and 260 m move in the same direction at 84 km/h and 48 km/h. Find the time taken by the faster train to cross the slower one.
Relative ##speed = 84 - 48 = 36## ##km / h = 10## m/s. Total ##distance = 140 + 260 = 400## m. ##Time = \dfrac{400}{10} = 40## s.
Q20. Two trains of lengths 220 m and 280 m cross each other in 25 s while moving in opposite directions. What is their relative speed in m/s?
Total ##distance = 220 + 280 = 500## m. Relative ##speed = \dfrac{500}{25} = 20## m/s.