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# Two cars are travelling in the same direction with a velocity of $60\,{{{\rm{km}}} {\left/ {\vphantom {{{\rm{km}}} {\rm{h}}}} \right.} {\rm{h}}}$ . They are separated by a distance of $5\,{\rm{km}}$. A truck moving in the opposite direction meets the two cars in a time interval of $3\min$. The velocity of the truck is (in${{{\rm{km}}} {\left/ {\vphantom {{{\rm{km}}} {\rm{h}}}} \right. } {\rm{h}}}$)​A. 20B. 30C. 40D. 60

Last updated date: 13th Sep 2024
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Hint: The velocity of a particle is defined as distance covered per unit time taken. When two bodies move in the opposite direction then their velocities add up to calculate the relative velocity and when two bodies are in the same direction then their velocities are subtracted to calculate the relative velocity.

Given: The velocity of two cars is$60\,{{{\rm{km}}} {\left/{\vphantom {{{\rm{km}}} {\rm{h}}}} \right.} {\rm{h}}}$. The distance between two cars and a truck is $5\,{\rm{km}}$. The time taken by the truck to meet cars is $3\,\min$.
The formula to calculate the relative velocity of the truck with respect to cars is ${v_t} = \dfrac{d}{t}$
Here, ${v_t}$ is the relative velocity of the truck, $d$ is the distance covered and $t$ is the time taken to cover the distance.
Substitute $5\,{\rm{km}}$ for$d$ and$3\,\min$ for $t$ in the formula to calculate the relative velocity of the truck.
${v_t} = \dfrac{{5\,{\rm{km}}}}{{\left( {3\,\min } \right)\left( {\dfrac{{1\,{\rm{h}}}}{{60\,\min }}} \right)}}\\ = \dfrac{{5\,{\rm{km}}}}{{0.05\,{\rm{h}}}}\\ = 100\,{{{\rm{km}}} {\left/ {\vphantom {{{\rm{km}}} {\rm{h}}}} \right. } {\rm{h}}}$
Since the truck moves in the opposite direction of the cars, thus the formula to calculate the relative velocity of the truck is
${v_t} = v + {v_c}$
Here, $v$ is the velocity of a truck and ${v_c}$ is the velocity of a car.
Substitute $100\,{{{\rm{km}}} {\left/ {\vphantom {{{\rm{km}}} {\rm{h}}}} \right. } {\rm{h}}}$ for ${v_t}$ and $60\,{{{\rm{km}}} {\left/ {\vphantom {{{\rm{km}}} {\rm{h}}}} \right. } {\rm{h}}}$ for ${v_c}$ in the formula and solve to calculate the velocity of the truck.
$100\,{{{\rm{km}}} {\left/ {\vphantom {{{\rm{km}}} {\rm{h}}}} \right. } {\rm{h}}} = v + 60{{{\rm{km}}} {\left/{\vphantom {{{\rm{km}}} {\rm{h}}}} \right. } {\rm{h}}}\\$
$\implies v = 100{{{\rm{km}}} {\left/{\vphantom {{{\rm{km}}} {\rm{h}}}} \right. } {\rm{h}}} -60{{{\rm{km}}} {\left/{\vphantom {{{\rm{km}}} {\rm{h}}}} \right. } {\rm{h}}}\\$
$= 40{{{\rm{km}}} {\left/{\vphantom {{{\rm{km}}} {\rm{h}}}} \right. } {\rm{h}}}$
Thus, the velocity of truck is $40{{{\rm{km}}} {\left/{\vphantom {{{\rm{km}}} {\rm{h}}}} \right.} {\rm{h}}}$

So, the correct answer is “Option C”.

Note:
The relative velocity of the system increases when the particles move in the opposite direction but it decreases when the particles move in the same direction.