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 1: Understand the concept of power
Power is defined as the rate at which work is done. The formula for power (P) is given by:
\[ P = \frac{W}{t} \]
where \( W \) is the work done and \( t \) is the time taken.

### Step 2: Identify the work done by each person
In this case, both persons A and B are lifting an object to the same height \( h \). The work done against gravity (potential energy gained) can be expressed as:
\[ W = mgh \]
where \( m \) is the mass of the object and \( g \) is the acceleration due to gravity.

### Step 3: Calculate the power for each person
For person A, who takes time \( t_1 = 2 \) seconds:
\[ P_A = \frac{W_A}{t_1} = \frac{mgh}{t_1} \]

For person B, who takes time \( t_2 = 4 \) seconds:
\[ P_B = \frac{W_B}{t_2} = \frac{mgh}{t_2} \]

### Step 4: Set up the ratio of powers
Now, we can find the ratio of their powers:
\[ \frac{P_A}{P_B} = \frac{\frac{mgh}{t_1}}{\frac{mgh}{t_2}} \]

### Step 5: Simplify the ratio
The \( mgh \) terms cancel out:
\[ \frac{P_A}{P_B} = \frac{t_2}{t_1} \]

### Step 6: Substitute the values of \( t_1 \) and \( t_2 \)
Substituting the given times:
\[ \frac{P_A}{P_B} = \frac{4}{2} = 2 \]

### Step 7: Write the final ratio
Thus, the ratio of their powers is:
\[ P_A : P_B = 2 : 1 \]

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

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