Problems on Trains

Convert speed to m/s: 54 km/hr = 54 * (5/18) = 15 m/s. Total distance covered in 30 seconds is 15 * 30 = 450 m. Subtracting the train length (90 m) gives the platform length: 450 - 90 = 360 m.

The time taken is (L1 + L2) / (V1 + V2). Using the provided notation, L1 + L2 is (L1 " L2) and V1 + V2 is (V1 " V2). Dividing these gives (L1 " L2) @ (V1 " V2).

Relative speed = Total distance / Time = (180 + 270) / 10.8 = 450 / 10.8 = 41.66 m/s. Convert to kmph: 41.66 * 18/5 = 150 kmph. Since they are in opposite directions, relative speed = S1 + S2. 150 = 60 + S2, so S2 = 90 kmph.

Relative speed = 54 - 46 = 8 km/hr = 8 * (5/18) m/s = 20/9 m/s. Total distance = 210 + 270 = 480 meters. Time = Distance / Speed = 480 / (20/9) = 480 * 9 / 20 = 24 * 9 = 216 seconds.

Relative speed = 80 - 40 = 40 m/s. Total distance to cover = 200 + 800 = 1000 m. Time = Distance / Relative Speed = 1000 / 40 = 25 seconds.

Since they start at the same time and meet, the time taken is the same. Ratio of speeds = Ratio of distances. One travels 110 km, the other 200-110 = 90 km. Ratio = 110:90 = 11:9.

The relative speed of the train with respect to the cart is distance / time = 130m / 8s = 16.25 m/s. Converting to km/hr: 16.25 * (18/5) = 58.5 km/hr. Since they move in the same direction, speed of train - speed of cart = 58.5, so speed of train = 58.5 + 4 = 62.5 km/hr.

Speed = 40 km/hr = 40 * (5/18) m/s = 100/9 m/s. Distance covered in 20s = (100/9) * 20 = 2000/9 = 222.22 m. Since distance = train length + bridge length, bridge length = 222.22 - 120 = 102.22 m.

Let speeds be v and 2v. Relative speed same direction = v. Relative speed opposite = 3v. Time = Distance / Relative Speed. 60 = D / v, so D = 60v. Time opposite = D / 3v = 60v / 3v = 20 seconds.

Speed = Length / Time. Statement I gives length. Statement II gives time to cross a pole (Speed = 240/24 = 10 m/s). Statement III is not needed.

Speed = 90 kmph = 90 * (5/18) m/s = 25 m/s. Total distance = Speed * Time = 25 * 9 = 225 m. Since the train length is 100 m, the platform length is 225 - 100 = 125 m.

Train speed = 60 km/h = 50/3 m/s. Man speed = 12 km/h = 10/3 m/s. Relative speed = 50/3 + 10/3 = 60/3 = 20 m/s. Train length = 20 * 6 = 120 m. Crossing platform: (120 + P) / (50/3) = 27. 120 + P = 27 * 50 / 3 = 450. P = 330 m.

Relative speed = 54 - 36 = 18 kmph = 5 m/s. Distance covered = 2 * length of train = 5 m/s * 60 s = 300 m. Length of each train = 300 / 2 = 150 m.

Let L be the length of the train and v be its speed. (L + 162) / 18 = v and (L + 120) / 15 = v. So, (L + 162) / 18 = (L + 120) / 15. 15(L + 162) = 18(L + 120). 15L + 2430 = 18L + 2160. 3L = 270, so L = 90.