If two person A and B take 2 s and 4 s, respectively to lift an object to the same height h, then the ratio of their powers is

Options

A. `1:2`
B. `1:1`
C. `2:1`
D. `1:3`

Explanation

To solve the problem of finding the ratio of powers of two persons A and B who lift an object to the same height in different times, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Power**:
Power is defined as the rate at which work is done. Mathematically, it is given by:
\[ P = \frac{W}{t} \]
where \( P \) is power, \( W \) is work done, and \( t \) is the time taken.

2. **Determine the Work Done**:
The work done in lifting an object to a height \( h \) against gravity is given by:
\[ W = mgh \]
where \( m \) is the mass of the object, \( g \) is the acceleration due to gravity, and \( h \) is the height.

3. **Calculate Power for Person A**:
Person A takes 2 seconds to lift the object. Therefore, the power \( P_A \) for person A can be calculated as:
\[ P_A = \frac{W}{t_A} = \frac{mgh}{2} \]

4. **Calculate Power for Person B**:
Person B takes 4 seconds to lift the same object. Therefore, the power \( P_B \) for person B can be calculated as:
\[ P_B = \frac{W}{t_B} = \frac{mgh}{4} \]

5. **Find the Ratio of Powers**:
Now, we can find the ratio of the powers of A and B:
\[ \frac{P_A}{P_B} = \frac{\frac{mgh}{2}}{\frac{mgh}{4}} \]
Simplifying this expression:
\[ \frac{P_A}{P_B} = \frac{mgh}{2} \times \frac{4}{mgh} = \frac{4}{2} = 2 \]

6. **Final Ratio**:
Therefore, the ratio of the powers of person A to person B is:
\[ P_A : P_B = 2 : 1 \]

### Conclusion:
The ratio of the powers of persons A and B is \( 2 : 1 \).