An engine draws a train up an incline of `1` in `100` at the rate `36km//h` If the resistance due to friction is `5kg` per metric ton calculate the power of the engine given mass of train and engine is `100` metic ton .

Explanation

To solve the problem, we need to calculate the power of the engine that draws a train up an incline. Here are the steps to find the solution:

### Step 1: Understanding the incline
The incline is given as `1 in 100`, which means for every 100 meters horizontally, the height increases by 1 meter. This gives us the sine of the angle θ:
\[ \sin \theta = \frac{1}{\sqrt{100^2 + 1^2}} = \frac{1}{\sqrt{10001}} \approx 0.01 \]

### Step 2: Convert speed from km/h to m/s
The speed of the train is given as `36 km/h`. To convert this to meters per second:
\[ \text{Speed} = \frac{36 \times 1000}{3600} = 10 \text{ m/s} \]

### Step 3: Calculate the weight of the train
The mass of the train and engine is given as `100 metric tons`. Converting this to kilograms:
\[ \text{Mass} = 100 \times 1000 = 100000 \text{ kg} \]

### Step 4: Calculate the gravitational force component along the incline
The force due to gravity acting down the incline is given by:
\[ F_{\text{gravity}} = mg \sin \theta \]
Where \( g \approx 9.8 \text{ m/s}^2 \):
\[ F_{\text{gravity}} = 100000 \times 9.8 \times \frac{1}{100} = 9800 \text{ N} \]

### Step 5: Calculate the frictional force
The resistance due to friction is given as `5 kg` per metric ton. Therefore, for `100 metric tons`:
\[ \text{Frictional force} = 5 \times 100 = 500 \text{ kg} \]
Converting this to Newtons:
\[ F_{\text{friction}} = 500 \times 9.8 = 4900 \text{ N} \]

### Step 6: Calculate the total force required
The total force required to move the train up the incline is the sum of the gravitational force and the frictional force:
\[ F_{\text{total}} = F_{\text{gravity}} + F_{\text{friction}} = 9800 + 4900 = 14700 \text{ N} \]

### Step 7: Calculate the power of the engine
Power is given by the formula:
\[ P = F \times v \]
Substituting the values we have:
\[ P = 14700 \times 10 = 147000 \text{ W} = 147 \text{ kW} \]

### Final Answer
The power of the engine is approximately **147 kW**.

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