Train Crossing Problems: Opposite vs Same Direction
This problem involves calculating the speed of trains based on crossing times when moving in opposite and same directions. We need to use the concept of relative speed.
Define Variables and Constants
- Length of Train A (\(L_A\)): 180 m
- Length of Train B (\(L_B\)): 220 m
- Total length for crossing (\(L_{total}\)): \(L_A + L_B = 180 + 220 = 400\) m
- Let the speeds of Train A and Train B be \(S_A\) m/s and \(S_B\) m/s, respectively.
- Assume \(S_A\) is the speed of the faster train.
Scenario 1: Trains Moving in Opposite Directions
When trains move in opposite directions, their relative speed is the sum of their individual speeds (\(S_A + S_B\)).
- Time taken (\(t_{opp}\)): 12 seconds
- Relative Speed = \(\frac{\text{Total Length}}{\text{Time taken}}\)
- \(S_A + S_B = \frac{400 \text{ m}}{12 \text{ s}} = \frac{100}{3}\) m/s. (Equation 1)
Scenario 2: Trains Moving in the Same Direction
When trains move in the same direction, their relative speed is the difference between their speeds (\(S_A - S_B\) for the faster train A).
- Time taken (\(t_{same}\)): 60 seconds
- Relative Speed = \(\frac{\text{Total Length}}{\text{Time taken}}\)
- \(S_A - S_B = \frac{400 \text{ m}}{60 \text{ s}} = \frac{20}{3}\) m/s. (Equation 2)
Solve for Speeds
We have a system of two linear equations:
- \(S_A + S_B = \frac{100}{3}\)
- \(S_A - S_B = \frac{20}{3}\)
Add Equation 1 and Equation 2:
\( (S_A + S_B) + (S_A - S_B) = \frac{100}{3} + \frac{20}{3} \) \( 2S_A = \frac{120}{3} \) \( 2S_A = 40 \) \( S_A = \frac{40}{2} = 20 \text{ m/s} \)Subtract Equation 2 from Equation 1:
\( (S_A + S_B) - (S_A - S_B) = \frac{100}{3} - \frac{20}{3} \) \( 2S_B = \frac{80}{3} \) \( S_B = \frac{40}{3} \text{ m/s} \)Since \(20 > \frac{40}{3}\), \(S_A\) is indeed the speed of the faster train.
Convert Speed to km/h
The speed of the faster train is \(S_A = 20\) m/s. To convert meters per second (m/s) to kilometers per hour (km/h), we multiply by \(\frac{18}{5}\).
\( \text{Speed in km/h} = 20 \times \frac{18}{5} \) \( \text{Speed in km/h} = 4 \times 18 \) \( \text{Speed in km/h} = 72 \)The speed of the faster train is 72 km/h.
Similar Questions
Two trains of equal length, running at the speeds of 60 km/h and 40 km/h, take 50 seconds to cross each other while they are running in the same direction. What time will they take to cross each other if they are running in opposite directions?
A 250 m long train running at a speed of 100 km/h crosses another train, coming from the opposite direction at a speed of 62 km/h, in 10 seconds. What is the length of the second train?
Two trains are running in opposite directions at the same speed. The length of each train is 175 m. If they cross each other in 7 s, the speed of each train is:
- A goods train and a superfast train started running from station A and station B, respectively, at the same time towards each other on parallel tracks. If the distance between station A and station B is 154 km and trains are running at speeds of 61 km/h and 93 km/h, respectively, then at how much distance (in km) from station B will the trains meet?
- Two trains of equal length are running on parallel lines in the same direction at speeds of 39 km/h and 21 km/h. The faster train passes the slower train in 26 seconds. The length of each train is:
- Two trains are moving in the same direction at speeds of 40 km/h and 20 km/h, respectively. The faster train completely overtakes a man seated in the slower train in 10 seconds. Find the length (in m) of the faster train. (Round off your answer to the nearest integer value in meters.)
- A faster train, 300 meters long, crosses a platform 600 meters in length in 40 seconds. A slower train, 450 meters long, takes 60 seconds to cross the same platform. Find the time it will take for the two trains to completely pass each other when traveling in opposite directions.
- A 628 m long train crosses a man walking at a speed of 10.1 km/h in the opposite direction in 12 seconds. What is the speed (in km/h) of the train?
- Train A is 260 m long and crosses a platform of length 160 m in 28 seconds. Train B is 340 m long and passes a stationary man in 20 seconds. Both trains travel at constant speeds on parallel straight tracks. Find the time (in seconds) taken when the faster train completely overtakes the slower train while moving in the same direction?
Two trains, each 250 m in length, are running on parallel lines in opposite directions at speeds of 90 km/h and 60 km/h respectively. In how many seconds will they cross each other completely?
Important Questions from Problem on Trains
Eight railway stations A, B, C, D, E, F, G and H are connected either by two-way passages or one-way passages. One-way passages are from C to A, E to G, B to F, D to H, G to C, E to C and H to G. Two-way passages are between A and E, G and B, F and D, and E and D.
If the route between G and C is closed, which one of the following stations need not be passed through while travelling from H to C?
A daily train is to be introduced between station A and station B starting from each end at 6 AM and the journey is to be completed in 42 hours. What is the number of trains needed in order to maintain the Shuttle Service?
A train with a uniform speed passes a 122 meters long platform in 17 seconds and a 210 meters long bridge in 25 seconds. The speed of the train is:
How long does a train 153 meters long running at the rate of 90 kmph take to cross a bridge 622 meters in length?
A train passes a 360 metre long platform in 40 seconds and a man standing on the platform in 16 seconds. The speed of the train is:
