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CAT 2023 Slot 2 — QA Question 10

Relative SpeedEasy

Ravi is driving at a speed of 40 km/h on a road. Vijay is 54 meters behind Ravi and driving in the same direction as Ravi. Ashok is driving along the same road from the opposite direction at a speed of 50 km/h and is 225 meters away from Ravi. The speed, in km/h, at which Vijay should drive so that all the three cross each other at the same time, is

Answer & solution

  • A

    67.2

  • B

    58.8

  • 61.6

  • D

    64.4

Solution

Easy

"All three cross at the same instant" means two separate meetings happen together: Ravi meets Ashok (head-on, so speeds add), and Vijay catches Ravi from behind (same direction, so speeds subtract). First find when Ravi and Ashok meet, then make Vijay close his 5454 m gap in exactly that time.

1

Convert speeds to m/s using ×518\times\tfrac{5}{18}:

Ravi=40×518=1009 m/sAshok=50×518=25018 m/s\begin{aligned} &\text{Ravi}=40\times\tfrac{5}{18}=\tfrac{100}{9}\ \text{m/s}\\ &\text{Ashok}=50\times\tfrac{5}{18}=\tfrac{250}{18}\ \text{m/s} \end{aligned}
2

Time for Ravi and Ashok to meet. They approach each other, so use the sum of speeds over the 225225 m gap:

t=2251009+25018=225200+25018=225×18450=9 s\begin{aligned} &t=\frac{225}{\tfrac{100}{9}+\tfrac{250}{18}}=\frac{225}{\tfrac{200+250}{18}}=\frac{225\times18}{450}=9\ \text{s} \end{aligned}
3

Vijay must catch Ravi in the same 99 s. He is 5454 m behind, closing at the difference of speeds (V−1009)\left(V-\tfrac{100}{9}\right) m/s:

9=54V−1009⇒ V−1009=6⇒ V=1549 m/s\begin{aligned} &9=\frac{54}{V-\tfrac{100}{9}}\\ &\Rightarrow\ V-\tfrac{100}{9}=6\\ &\Rightarrow\ V=\tfrac{154}{9}\ \text{m/s} \end{aligned}
4

Back to km/h using ×185\times\tfrac{18}{5}:

V=1549×185=154×25=61.6 km/h\begin{aligned} &V=\tfrac{154}{9}\times\tfrac{18}{5}=\tfrac{154\times2}{5}=61.6\ \text{km/h} \end{aligned}

Vijay should drive at 61.6\mathbf{61.6} km/h.

CAT 2023 Slot 2 QA Q10: Ravi is driving at a speed of 40 km/h on a road. Vijay is 54 meters behind Ravi and driving in the same direct — Solution | TheCATExam