A 60 kg man runs up a staircase in 12 seconds while 50 kg man runs up the same staircase in 11, seconds, the ratio of the rate of doing their work is

Options

A. `6:5`
B. `12:11`
C. `11:10`
D. 10:11`

Explanation

To solve the problem of finding the ratio of the rate of doing work (power) between two men running up a staircase, we can follow these steps:

### Step 1: Understand the Work Done
The work done by each man while running up the staircase is equal to the gain in potential energy, which can be expressed as:
\[ \text{Work} = mgh \]
where:
- \( m \) = mass of the man
- \( g \) = acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \))
- \( h \) = height of the staircase

### Step 2: Define Power
Power is defined as the rate of doing work, which can be expressed as:
\[ \text{Power} = \frac{\text{Work}}{\text{Time}} \]
Substituting the expression for work, we get:
\[ \text{Power} = \frac{mgh}{t} \]
where:
- \( t \) = time taken to run up the staircase

### Step 3: Set Up the Ratios
Let:
- \( P_1 \) = Power of the 60 kg man
- \( P_2 \) = Power of the 50 kg man

Using the formula for power:
\[ P_1 = \frac{m_1gh}{t_1} \]
\[ P_2 = \frac{m_2gh}{t_2} \]

### Step 4: Find the Ratio of Powers
Now, we can find the ratio of their powers:
\[ \frac{P_1}{P_2} = \frac{\frac{m_1gh}{t_1}}{\frac{m_2gh}{t_2}} \]

### Step 5: Simplify the Ratio
The \( gh \) terms cancel out:
\[ \frac{P_1}{P_2} = \frac{m_1}{m_2} \cdot \frac{t_2}{t_1} \]

### Step 6: Substitute the Values
Substituting the values:
- \( m_1 = 60 \, \text{kg} \)
- \( m_2 = 50 \, \text{kg} \)
- \( t_1 = 12 \, \text{s} \)
- \( t_2 = 11 \, \text{s} \)

We get:
\[ \frac{P_1}{P_2} = \frac{60}{50} \cdot \frac{11}{12} \]

### Step 7: Calculate the Ratio
Now, calculate the ratio:
\[ \frac{P_1}{P_2} = \frac{60 \cdot 11}{50 \cdot 12} \]
\[ = \frac{660}{600} \]
\[ = \frac{11}{10} \]

### Conclusion
Thus, the ratio of the rate of doing work (power) of the two men is:
\[ \frac{P_1}{P_2} = \frac{11}{10} \]

### Final Answer
The correct option is \( 11:10 \).

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