CAT 2023 Slot 2 QA Question 11

Consider the scenario. Ravi is driving at 40 km/h. Vijay is behind Ravi, 54 meters away, going the same direction. Ashok is coming from the opposite direction, 225 meters away from Ravi, driving at 50 km/h. We need all three to cross each other at the same time, which I think means they all meet at the same point at the same time.

Wait, but how can three people cross each other at the same time? I guess it means that when Ashok meets Ravi, Vijay should also meet Ravi at the same point. Or maybe all three meet each other pairwise at the same time. Hmm, maybe I need to clarify that.

Since Ashok is coming from the opposite direction, he will meet Ravi after some time. At the same time, Vijay, who is behind Ravi, needs to reach the same point where Ashok and Ravi meet. So, the time it takes for Ashok to meet Ravi should be the same as the time it takes for Vijay to cover the 54 meters gap and reach Ravi's starting point, but wait, no, because Ravi is moving too.

Wait, maybe I should model their positions as functions of time. Let's denote the time when they all meet as 't' hours. Let me convert all speeds to m/s because the distances are given in meters.

So, Ravi's speed is 40 km/h. To convert to m/s, I multiply by 1000/3600, which is 5/9. So, 40 * (5/9) = 200/9 ≈ 22.222 m/s.

Vijay's speed is what we need to find, let's call it 'v' km/h. Converting that to m/s, it's v * (5/9) m/s.

Ashok is coming from the opposite direction at 50 km/h. So, his speed in m/s is 50 * (5/9) ≈ 27.778 m/s.

Now, let's think about the positions. Let me assume that at time t=0, Ravi is at position 0. So, Vijay is at position -54 meters (since he's 54 meters behind). Ashok is at position +225 meters (since he's 225 meters away from Ravi in the opposite direction).

if Ravi is at position 0, and Ashok is 225 meters away from Ravi but coming from the opposite direction, so Ashok's starting position is +225 meters. Vijay is behind Ravi, so his starting position is -54 meters.

Now, after time 't' hours, Ravi would have moved a distance of (40 km/h * t) converted to meters. Wait, but I converted speeds to m/s earlier, so maybe it's better to keep time in seconds. Wait, no, let me think again.

Wait, maybe it's better to keep time in hours since the speeds are in km/h. So, in time 't' hours, Ravi would have moved 40t km, which is 40t * 1000 meters = 40,000t meters.

Similarly, Ashok, moving towards Ravi at 50 km/h, would cover 50t km, which is 50,000t meters.

But wait, the initial distance between Ashok and Ravi is 225 meters. So, the time it takes for Ashok and Ravi to meet is when the sum of the distances they've covered equals 225 meters.

Wait, no, because they're moving towards each other, their relative speed is the sum of their speeds. So, the time taken for them to meet is the initial distance divided by their relative speed.

Wait, but I converted speeds to m/s earlier, so maybe I should work in m/s to keep units consistent.

Let me try that approach again.

Ravi's speed: 40 km/h = 40 (1000/3600) = 40 (5/18) ≈ 11.111 m/s.

Ashok's speed: 50 km/h = 50 * (5/18) ≈ 13.889 m/s.

Since they're moving towards each other, their relative speed is 11.111 + 13.889 = 25 m/s.

The initial distance between them is 225 meters, so the time until they meet is 225 / 25 = 9 seconds.

So, t = 9 seconds.

Now, in these 9 seconds, Ravi would have moved a distance of 11.111 m/s * 9 s ≈ 100 meters.

Similarly, Ashok would have moved 13.889 m/s * 9 s ≈ 125 meters, which adds up to 225 meters, so that checks out.

Now, at the same time, Vijay needs to reach the same point where Ravi and Ashok meet. So, where is Ravi after 9 seconds? He's at 100 meters from his starting point.

But wait, Vijay is starting 54 meters behind Ravi, so his starting position is -54 meters. He needs to reach the point where Ravi is after 9 seconds, which is +100 meters from the starting point.

Wait, no, that can't be right because Ravi is moving forward, so after 9 seconds, Ravi is at +100 meters from his starting point. So, Vijay needs to cover the distance from -54 meters to +100 meters, which is a total of 154 meters in 9 seconds.

Wait, but that seems like a lot. Let me check.

if Ravi is moving forward, then after 9 seconds, Ravi is at position 100 meters. So, Vijay needs to reach that same point at the same time, starting from -54 meters.

So, the distance Vijay needs to cover is 100 - (-54) = 154 meters in 9 seconds.

So, Vijay's speed in m/s is 154 meters / 9 seconds ≈ 17.111 m/s.

Now, converting that back to km/h, we multiply by 18/5, so 1