Train Lengths and Variables
Given the lengths of the two trains: $L_1 = 200$ m and $L_2 = 400$ m. Let their speeds be $S_1$ and $S_2$ (in m/s), with $S_1$ being the higher speed.
The total distance $D$ the trains cover relative to each other is the sum of their lengths:
$D = L_1 + L_2 = 200 \text{ m} + 400 \text{ m} = 600 \text{ m}$Relative Speed Calculations
1. Same Direction:
When traveling in the same direction, the relative speed is $S_{rel\_same} = S_1 - S_2$. The time taken to overtake is $t_{same} = 30$ seconds.
Applying the distance formula ($D = \text{speed} \times \text{time}$):
$600 \text{ m} = (S_1 - S_2) \times 30 \text{ s}$Calculating the relative speed:
$S_1 - S_2 = \frac{600}{30} \text{ m/s} = 20 \text{ m/s}$2. Opposite Directions:
When traveling in opposite directions, the relative speed is $S_{rel\_opp} = S_1 + S_2$. The time taken to cross is $t_{opp} = 6$ seconds.
Applying the distance formula:
$600 \text{ m} = (S_1 + S_2) \times 6 \text{ s}$Calculating the relative speed:
$S_1 + S_2 = \frac{600}{6} \text{ m/s} = 100 \text{ m/s}$Solving for Speeds in m/s
We have the following system of equations:
- Equation 1: $S_1 - S_2 = 20$ m/s
- Equation 2: $S_1 + S_2 = 100$ m/s
Adding Equation 1 and Equation 2:
$(S_1 - S_2) + (S_1 + S_2) = 20 + 100$ $2S_1 = 120 \text{ m/s}$ $S_1 = 60 \text{ m/s}$Substituting $S_1 = 60$ m/s into Equation 2:
$60 \text{ m/s} + S_2 = 100 \text{ m/s}$ $S_2 = 40 \text{ m/s}$Converting Speeds to km/h
To convert speeds from meters per second (m/s) to kilometers per hour (km/h), multiply by the conversion factor $\frac{18}{5}$.
Higher speed ($S_1$):
$S_1 = 60 \times \frac{18}{5} \text{ km/h} = 12 \times 18 \text{ km/h} = 216 \text{ km/h}$Lower speed ($S_2$):
$S_2 = 40 \times \frac{18}{5} \text{ km/h} = 8 \times 18 \text{ km/h} = 144 \text{ km/h}$Resulting Train Speeds
The speeds of the two trains are 216 km/h and 144 km/h.
Similar Questions
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Important Questions from Relative Speed
In a circular race of 2500 m, a man and a woman start from a point towards opposite directions with speeds of 37 km/h and 35 km/h, respectively. After how much time from the start of the race will they meet for the first time?
- X and Yrun a 3 km race along a circular course of length 300 m. Their speeds are in the ratio 3 : 2. If they start together in the same direction, how many times would the first one pass the other (the start-off is not counted as passing)?
A train of length 600 meters passes another train of length 1000 meters moving in the opposite direction in 96 seconds. If the speed of both the trains is the same, then what is the speed of each train?
The distance between Delhi and Patna is $480$ km. A train starts from Delhi and travels towards Patna at a speed of $60$ kmph. Another train starts from Patna towards Delhi at a speed of $80$ kmph, but starts $1$ hour after the first train. In how much time, after the first train started, will they meet each other?
Ram traveling at the speed of 3 km/h reaches 15 minutes late. Had he walked at 4 km/h, he would have reached 15 minutes earlier. How much distance does Ram have to cover?
