Train P crosses a pole in $32$ seconds and takes $72$ seconds to cross a platform that is $1200$ meters long. Train Q has a speed of $72 ext{ km/h}$ and a length of $1280$ meters. Both trains run in the same direction. What is the time taken by train P to cross train Q?

Question

Train P crosses a pole in $32$ seconds and takes $72$ seconds to cross a platform that is $1200$ meters long. Train Q has a speed of $72 	ext{ km/h}$ and a length of $1280$ meters. Both trains run in the same direction. What is the time taken by train P to cross train Q?

Accepted Answer

**Solution:**

1. First, find the length of train P.
   - Train P crosses a pole in $32$ seconds, so its speed is:
     $$	ext{Speed of P} = \frac{	ext{Length of P}}{32}$$

2. Train P crosses a platform of length $1200$ meters in $72$ seconds.
   - Total distance covered while crossing the platform = Length of train P + Length of platform = $L_P + 1200$
   - Speed of train P = $$\frac{L_P + 1200}{72}$$

3. Since speed of train P is the same in both cases:
   $$\frac{L_P}{32} = \frac{L_P + 1200}{72}$$

4. Cross-multiply and solve for $L_P$:
   $$72 L_P = 32 (L_P + 1200)$$
   $$72 L_P = 32 L_P + 38400$$
   $$72 L_P - 32 L_P = 38400$$
   $$40 L_P = 38400$$
   $$L_P = \frac{38400}{40} = 960 	ext{ meters}$$

5. Calculate speed of train P:
   $$	ext{Speed of P} = \frac{L_P}{32} = \frac{960}{32} = 30 	ext{ m/s}$$

6. Convert speed of train Q from km/h to m/s:
   $$72 	ext{ km/h} = \frac{72 	imes 1000}{3600} = 20 	ext{ m/s}$$

7. Since both trains move in the same direction, relative speed = speed of P - speed of Q = $30 - 20 = 10 	ext{ m/s}$

8. Total length to be crossed when train P crosses train Q = Length of P + Length of Q = $960 + 1280 = 2240$ meters

9. Time taken to cross each other:
   $$	ext{Time} = \frac{	ext{Total length}}{	ext{Relative speed}} = \frac{2240}{10} = 224 	ext{ seconds}$$

**Answer:** $224 	ext{ seconds}$

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