A car of mass 1500 kg is lifted up a distance of 30 m by crane A in 0.5 minutes. The second crane B does the same jod in 1 minute . The ratio of their powers is

Options

A. `1:2`
B. `2:1`
C. `1:4`
D. `4:1`

Answer

To find the ratio of powers of the two cranes lifting the car, we can follow these steps:

### Step 1: Understand the power formula
Power is defined as the work done per unit time. 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: Calculate the work done
The work done in lifting the car can be calculated using the formula:
\[ W = mgh \]
where:
- \( m \) = mass of the car = 1500 kg
- \( g \) = acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \))
- \( h \) = height lifted = 30 m

Substituting the values:
\[ W = 1500 \times 9.81 \times 30 \]
Calculating this gives:
\[ W = 1500 \times 9.81 \times 30 = 441450 \, \text{Joules} \]

### Step 3: Calculate the power for Crane A
Crane A lifts the car in 0.5 minutes. We need to convert this time into seconds:
\[ t_A = 0.5 \, \text{minutes} = 0.5 \times 60 = 30 \, \text{seconds} \]
Now, we can calculate the power for Crane A:
\[ P_A = \frac{W}{t_A} = \frac{441450}{30} \]
Calculating this gives:
\[ P_A = 14715 \, \text{Watts} \]

### Step 4: Calculate the power for Crane B
Crane B lifts the car in 1 minute. Again, we convert this time into seconds:
\[ t_B = 1 \, \text{minute} = 1 \times 60 = 60 \, \text{seconds} \]
Now, we can calculate the power for Crane B:
\[ P_B = \frac{W}{t_B} = \frac{441450}{60} \]
Calculating this gives:
\[ P_B = 7374.5 \, \text{Watts} \]

### Step 5: Find the ratio of powers
Now, we can find the ratio of the powers of Crane A to Crane B:
\[ \text{Ratio} = \frac{P_A}{P_B} = \frac{14715}{7374.5} \]
Calculating this gives:
\[ \text{Ratio} \approx 2 \]
Thus, the ratio of their powers is:
\[ \text{Ratio of powers} = 2:1 \]

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