Q129 / 180 · COMEDK 2024 (Morning Shift) · Physics
When two objects are moving along a straight line in the same direction, the distance between them increases by $6 \mathrm{~m}$ in one second. If the objects move with their constant speed towards each other the distance decreases by $8 \mathrm{~m}$ in one second, then the speed of the objects are :
- A$14 \mathrm{~ms}^{-1}$ and $2 \mathrm{~ms}^{-1}$
- Correct answerB$7 \mathrm{~ms}^{-1}$ and $1 \mathrm{~ms}^{-1}$
- C$3.5 \mathrm{~ms}^{-1}$ and $1 \mathrm{~ms}^{-1}$
- D$3.5 \mathrm{~ms}^{-1}$ and $2 \mathrm{~ms}^{-1}$
Solution
Let the speeds of the two objects be $v_1$ and $v_2$ in $\text{ms}^{-1}$, where $v_1 > v_2$.
When moving in the same direction, the relative speed is $v_1 - v_2$. The distance between them increases by $6 \text{ m}$ in $1 \text{ s}$, so $v_1 - v_2 = \dfrac{6 \text{ m}}{1 \text{ s}} = 6 \text{ ms}^{-1}$.
When moving towards each other, the relative speed is $v_1 + v_2$. The distance between them decreases by $8 \text{ m}$ in $1 \text{ s}$, so $v_1 + v_2 = \dfrac{8 \text{ m}}{1 \text{ s}} = 8 \text{ ms}^{-1}$.
Adding the two equations: $(v_1 - v_2) + (v_1 + v_2) = 6 + 8 \Rightarrow 2v_1 = 14 \Rightarrow v_1 = 7 \text{ ms}^{-1}$.
Subtracting the first equation from the second: $(v_1 + v_2) - (v_1 - v_2) = 8 - 6 \Rightarrow 2v_2 = 2 \Rightarrow v_2 = 1 \text{ ms}^{-1}$.
The speeds of the objects are $7 \text{ ms}^{-1}$ and $1 \text{ ms}^{-1}$.
Answer: $7 \mathrm{~ms}^{-1}$ and $1 \mathrm{~ms}^{-1}$
When moving in the same direction, the relative speed is $v_1 - v_2$. The distance between them increases by $6 \text{ m}$ in $1 \text{ s}$, so $v_1 - v_2 = \dfrac{6 \text{ m}}{1 \text{ s}} = 6 \text{ ms}^{-1}$.
When moving towards each other, the relative speed is $v_1 + v_2$. The distance between them decreases by $8 \text{ m}$ in $1 \text{ s}$, so $v_1 + v_2 = \dfrac{8 \text{ m}}{1 \text{ s}} = 8 \text{ ms}^{-1}$.
Adding the two equations: $(v_1 - v_2) + (v_1 + v_2) = 6 + 8 \Rightarrow 2v_1 = 14 \Rightarrow v_1 = 7 \text{ ms}^{-1}$.
Subtracting the first equation from the second: $(v_1 + v_2) - (v_1 - v_2) = 8 - 6 \Rightarrow 2v_2 = 2 \Rightarrow v_2 = 1 \text{ ms}^{-1}$.
The speeds of the objects are $7 \text{ ms}^{-1}$ and $1 \text{ ms}^{-1}$.
Answer: $7 \mathrm{~ms}^{-1}$ and $1 \mathrm{~ms}^{-1}$
Hence, the correct answer is option: 'B'.