Train Meeting Problem Solver

The Train Meeting Problem Solver tackles the five most common train word problems in one place: two trains meeting head-on, a faster train overtaking a slower one in the same direction, two trains of given lengths crossing each other, a train passing a single point such as a pole or signal post, and a train crossing a platform, bridge, or tunnel. Type in the speeds, distances, and lengths in any mix of metric (km/h, m/s, km, m) or imperial units (mph, ft/s, mi, ft) — the solver converts everything to consistent SI units, applies the right relative-speed rule, and shows a full LaTeX-formatted solution alongside an animated track that visualizes the motion in real proportion.

How to use this solver

1. Pick the scenario that matches your problem from the dropdown — meeting, overtaking, two crossing, passing a pole, or crossing a platform.
2. Choose the display units. You can still mix km/h with metres, or mph with feet — the solver handles unit conversion internally.
3. Enter the speeds, lengths, and gap. For the meeting scenario you can optionally add a head-start time for Train A in minutes.
4. Click Solve. The headline value shows the meeting, overtaking, or crossing time. Below it you get the relative speed, distance covered by each train, and a step-by-step LaTeX explanation.
5. Watch the animated track at the side and in the result panel — the trains move in real proportion to the speeds and direction of motion.

The five formulas at a glance

The relative-speed rule (the key idea)

Almost every train word problem reduces to one identity:

\[ \text{time} \;=\; \dfrac{\text{distance to cover}}{\text{relative speed}} \]

What changes between scenarios is the meaning of "distance" and the sign of the relative speed:

• Toward each other — the two trains close the gap together, so add their speeds: \( v_{rel} = v_1 + v_2 \).
• Same direction — only the difference in speed closes the gap: \( v_{rel} = v_B - v_A \). If both speeds are equal, the gap never closes.
• Crossing with lengths — the back of one train must clear the back of the other, so the distance to cover equals \( L_1 + L_2 \) instead of just the gap.
• Pole vs platform — a pole is a point (cover \( L_{train} \)); a platform has length (cover \( L_{train} + L_{platform} \)).

Worked example: meeting head-on

Two trains start 300 km apart and move toward each other. Train A travels at 60 km/h, Train B at 90 km/h. When and where do they meet?

1. Convert speeds to m/s: \( v_1 = 60 \times \tfrac{5}{18} = 16.667 \) m/s; \( v_2 = 90 \times \tfrac{5}{18} = 25 \) m/s.
2. Relative speed: \( v_{rel} = 16.667 + 25 = 41.667 \) m/s = 150 km/h.
3. Time to meet: \( t = \dfrac{300\,000\;\text{m}}{41.667\;\text{m/s}} = 7200 \) s = 2 h.
4. Distance covered by Train A: \( d_1 = 60 \times 2 = 120 \) km, so the meeting point is 120 km from A and 180 km from B.

Worked example: train crossing a platform

A 150 m train moving at 90 km/h needs to cross a 350 m platform.

1. Convert the speed: \( 90 \times \tfrac{5}{18} = 25 \) m/s.
2. Total distance to cover: \( 150 + 350 = 500 \) m.
3. Crossing time: \( t = \dfrac{500}{25} = 20 \) s.

Common pitfalls and how to avoid them

• Mixing units — multiplying km/h by seconds gives a number that means nothing. Either convert km/h to m/s by multiplying by \(\tfrac{5}{18}\), or convert m/s to km/h by multiplying by 3.6. The solver does this automatically.
• Forgetting train length — when two trains cross or a train crosses a platform, the back of the train must clear the other end. Always add lengths to the distance.
• Wrong sign on relative speed — if you write \( v_1 + v_2 \) for a same-direction overtake, the time will be far too short. Add only when motion is opposing.
• Equal speeds in same direction — if the two trains have equal speed and travel the same way, the relative speed is zero and they never overtake.
• Head start vs distance — a head start is a time advantage, not a distance. Convert it to distance by multiplying the leader's speed by the head-start time.

Quick conversion reference

Frequently asked questions