When two bodies approach each other with the different speeds, the distance between them decreases by 120 m for every one minute. If they are moving in direction, the distance between them increases by 90 m for very one minute. The speeds of the bodies are

Options

A. `2ms^(-1)and0.5ms^(-1)`
B. `3ms^(-1)and2ms^(-1)`
C. `1.75ms^(-1)and0.25ms^(-1)`
D. `2.5ms^(-1)and0.5ms^(-1)`

Answer

To solve the problem, we need to find the speeds of two bodies based on their relative motion. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We have two cases:
1. When the two bodies approach each other, the distance between them decreases by 120 m in one minute.
2. When they move in the same direction, the distance between them increases by 90 m in one minute.

### Step 2: Set Up the Equations
Let:
- \( v_1 \) = speed of the first body (in m/s)
- \( v_2 \) = speed of the second body (in m/s)

**Case 1: Approaching Each Other**
When the bodies are approaching each other, their speeds add up. The equation for this case is:
\[ v_1 + v_2 = \frac{120 \text{ m}}{1 \text{ min}} = \frac{120 \text{ m}}{60 \text{ s}} = 2 \text{ m/s} \]

**Case 2: Moving in the Same Direction**
When the bodies are moving in the same direction, the relative speed is the difference of their speeds. The equation for this case is:
\[ v_1 - v_2 = \frac{90 \text{ m}}{1 \text{ min}} = \frac{90 \text{ m}}{60 \text{ s}} = 1.5 \text{ m/s} \]

### Step 3: Solve the System of Equations
Now we have a system of two equations:
1. \( v_1 + v_2 = 2 \) (Equation 1)
2. \( v_1 - v_2 = 1.5 \) (Equation 2)

We can solve these equations by adding them together:
\[ (v_1 + v_2) + (v_1 - v_2) = 2 + 1.5 \]
This simplifies to:
\[ 2v_1 = 3.5 \implies v_1 = \frac{3.5}{2} = 1.75 \text{ m/s} \]

### Step 4: Find \( v_2 \)
Now, substitute \( v_1 \) back into Equation 1 to find \( v_2 \):
\[ 1.75 + v_2 = 2 \implies v_2 = 2 - 1.75 = 0.25 \text{ m/s} \]

### Step 5: Conclusion
The speeds of the two bodies are:
- \( v_1 = 1.75 \text{ m/s} \)
- \( v_2 = 0.25 \text{ m/s} \)

### Final Answer
The speeds of the bodies are 1.75 m/s and 0.25 m/s.

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