An engine develops 10 kW of power. How much time will it take to lift a mass of 200 kg to a height of 40 m (`g=10(m)/(sec^2)`)`?

Options

A. 4 sec
B. 5 sec
C. 8 sec
D. 10 sec

Explanation

To solve the problem of how much time it will take for an engine that develops 10 kW of power to lift a mass of 200 kg to a height of 40 m, we can follow these steps:

### Step 1: Understand the Given Data
- Power (P) = 10 kW = \(10 \times 10^3\) W = 10,000 W
- Mass (m) = 200 kg
- Height (h) = 40 m
- Gravitational acceleration (g) = 10 m/s²

### Step 2: Calculate the Potential Energy (PE)
The potential energy gained by lifting a mass to a height is given by the formula:
\[ PE = mgh \]
Substituting the values:
\[ PE = 200 \, \text{kg} \times 10 \, \text{m/s}^2 \times 40 \, \text{m} \]
Calculating this:
\[ PE = 200 \times 10 \times 40 = 80,000 \, \text{J} \]

### Step 3: Relate Power to Work Done and Time
Power is defined as the rate of doing work, which can also be expressed in terms of energy:
\[ P = \frac{W}{t} \]
Where \(W\) is the work done (which is equal to the potential energy in this case) and \(t\) is the time taken. Rearranging this gives:
\[ t = \frac{W}{P} \]

### Step 4: Substitute the Values into the Equation
Now substitute the values of work done (which is the potential energy) and power into the equation:
\[ t = \frac{PE}{P} = \frac{80,000 \, \text{J}}{10,000 \, \text{W}} \]
Calculating this gives:
\[ t = \frac{80,000}{10,000} = 8 \, \text{s} \]

### Conclusion
The time taken to lift the mass is **8 seconds**.

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