To find the speed of the train, we can approach the problem step-by-step. Let the speed of the train be $V$ km/h, and the length of the train be $L$ meters.

Key information:

• The speeds of the two persons are 3 km/h and 5 km/h.
• The train crosses the first person in 10 seconds and the second person in 11 seconds.
• We need to convert the speeds to meters per second since the time is given in seconds.

Step 1: Convert speeds to meters per second

1 km/h = $\frac{1000}{3600}$ m/s, so:

• Speed of the first person = $3 \times \frac{1000}{3600} = \frac{5}{6}$ m/s.
• Speed of the second person = $5 \times \frac{1000}{3600} = \frac{5}{3}$ m/s.

Step 2: Relative speed of the train with respect to each person

• For the first person, relative speed = $(V - 3)$ km/h.
• For the second person, relative speed = $(V - 5)$ km/h.

Convert these relative speeds to meters per second:

• Relative speed with respect to the first person: $(V - 3) \times \frac{1000}{3600}$ m/s.
• Relative speed with respect to the second person: $(V - 5) \times \frac{1000}{3600}$ m/s.

Step 3: Use the time to cross the two persons

• The train crosses the first person in 10 seconds, so the length of the train $L$ = relative speed × time: $L = \left( \frac{(V - 3) \times 1000}{3600} \right) \times 10.$
• The train crosses the second person in 11 seconds, so: $L = \left( \frac{(V - 5) \times 1000}{3600} \right) \times 11.$

Step 4: Equate the two expressions for $L$

$\frac{(V - 3) \times 1000}{3600} \times 10 = \frac{(V - 5) \times 1000}{3600} \times 11.$ Simplifying: $(V - 3) \times 10 = (V - 5) \times 11.$

Expanding both sides: $10V - 30 = 11V - 55.$

Solving for $V$: $55 - 30 = 11V - 10V,$ $V = 25 \text{ km/h}.$

Thus, the speed of the train is 25 km/h.

Would you like more details or have any questions?

Here are 5 related questions:

1. How would the time change if the train's speed were increased by 10%?
2. What if the people were moving in the opposite direction?
3. How would the length of the train impact the solution?
4. Can you find the length of the train using this method?
5. How can we calculate the speed if more people were involved?

Tip: Always ensure that units are consistent when solving such problems—convert km/h to m/s when time is given in seconds.