A man `M_(1)` of mass `80 kg` runs up a staircase in `15 s`. Another man `M_(2)` also of mass `80 kg` runs up the same staircase in `20 s`. The ratio of the power development by then will be:

Options

A. (a) `1`
B. (b) `4/3`
C. (c) `16/9`
D. (d) None of the above

Explanation

To find the ratio of the power developed by two men running up a staircase, we can follow these steps:

### Step 1: Understand the concept of power
Power is defined as the rate at which work is done. Mathematically, it can be expressed as:
\[ P = \frac{W}{t} \]
where \( P \) is power, \( W \) is work done, and \( t \) is the time taken.

### Step 2: Identify the work done by both men
Since both men have the same mass and are running up the same staircase, the work done \( W \) by both men will be the same. The work done against gravity when moving to a height \( h \) is given by:
\[ W = mgh \]
where \( m \) is mass, \( g \) is acceleration due to gravity, and \( h \) is the height of the staircase.

### Step 3: Calculate the power for each man
Let:
- \( M_1 \) (man 1) runs up the staircase in \( t_1 = 15 \) seconds.
- \( M_2 \) (man 2) runs up the staircase in \( t_2 = 20 \) seconds.

The power developed by man 1 is:
\[ P_1 = \frac{W}{t_1} = \frac{W}{15} \]

The power developed by man 2 is:
\[ P_2 = \frac{W}{t_2} = \frac{W}{20} \]

### Step 4: Find the ratio of the powers
To find the ratio of the powers developed by the two men, we can write:
\[ \frac{P_1}{P_2} = \frac{\frac{W}{15}}{\frac{W}{20}} \]

Since \( W \) is the same for both, it cancels out:
\[ \frac{P_1}{P_2} = \frac{20}{15} \]

### Step 5: Simplify the ratio
Now, simplifying the ratio:
\[ \frac{P_1}{P_2} = \frac{20}{15} = \frac{4}{3} \]

### Conclusion
Thus, the ratio of the power developed by man 1 to man 2 is:
\[ P_1 : P_2 = 4 : 3 \]

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