This problem involves calculating the change in distance between two individuals, Raju and Karthik, who are moving at different speeds. Since Karthik is chasing Raju, they are running in the same direction, and we can use the concept of relative speed.
Given Information
- Initial distance between Raju and Karthik: 300 m
- Raju's speed ($v_R$): 10 km/h
- Karthik's speed ($v_K$): 12 km/h
- Time duration ($t$): 6 minutes
Convert Speeds to Consistent Units
To perform calculations, convert speeds from km/h to meters per minute (m/min).
- Raju's speed: $v_R = 10 \text{ km/h} = 10 \times \frac{1000 \text{ m}}{60 \text{ min}} = \frac{10000}{60} \text{ m/min} = \frac{500}{3} \text{ m/min}$
- Karthik's speed: $v_K = 12 \text{ km/h} = 12 \times \frac{1000 \text{ m}}{60 \text{ min}} = \frac{12000}{60} \text{ m/min} = 200 \text{ m/min} $
Calculate Relative Speed
Since Karthik is chasing Raju (running in the same direction) and Karthik is faster, the relative speed at which Karthik closes the distance is the difference between their speeds.
Relative speed ($v_{rel}$) = Karthik's speed - Raju's speed
$v_{rel} = v_K - v_R$ $v_{rel} = 200 \text{ m/min} - \frac{500}{3} \text{ m/min}$ $v_{rel} = \frac{600 - 500}{3} \text{ m/min} = \frac{100}{3} \text{ m/min}$Calculate Distance Closed
Calculate the reduction in distance between them after 6 minutes using the relative speed.
Distance closed = Relative speed $\times$ Time
$ \text{Distance closed} = v_{rel} \times t $ $ \text{Distance closed} = \frac{100}{3} \text{ m/min} \times 6 \text{ min} $ $ \text{Distance closed} = 100 \times 2 \text{ m} = 200 \text{ m} $Determine Final Distance
The initial distance is reduced by the distance closed.
Final distance = Initial distance - Distance closed
$ \text{Final distance} = 300 \text{ m} - 200 \text{ m} $ $ \text{Final distance} = 100 \text{ m} $Therefore, the distance between Raju and Karthik after 6 minutes is 100 m.
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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?
