An engine approaches a hill with a constant speed. When it is at a distance of 0.9 km, it blows a whistle whose echo is heard by the driver after 5 seconds. If the speed of sound in air is 330 m/s, then the speed of the engine is :

An engine approaches a hill with a constant speed. The whistle is blown when the engine is 0.9 km away from the hill. Convert this distance to meters: 0.9 km = 900 meters. The echo is heard by the driver after 5 seconds, and the speed of sound is 330 m/s. We need to find the speed of the engine, denoted as $$ v $$ m/s.

When the whistle is blown, the sound travels towards the hill at 330 m/s. The hill is stationary. The echo is the sound reflecting off the hill and returning to the engine. During the 5 seconds, the engine is moving towards the hill, so the distance covered by the sound includes the initial distance to the hill and the reduced distance on the return due to the engine's movement.

Consider the total distance traveled by the sound in 5 seconds. Since the speed of sound is 330 m/s, the total distance covered by sound is:

$$ \text{Distance} = \text{Speed} \times \text{Time} = 330 \times 5 = 1650 \text{ meters}. $$

This distance is the sum of the distance from the engine's initial position to the hill and the distance from the hill back to the engine's position when the echo is heard.

Let the initial position of the engine be point A, and the hill be point B. So, AB = 900 meters. In 5 seconds, the engine moves towards the hill with speed $$ v $$ m/s, so the distance covered by the engine is $$ v \times 5 $$ meters. Let the position of the engine when the echo is heard be point D. Therefore, AD = $$ 5v $$ meters.

The distance from D to the hill (point B) is DB. Since D is between A and B, DB = AB - AD = 900 - 5v meters.

The sound travels from A to B (900 meters) and then from B to D (DB = 900 - 5v meters). So, the total distance traveled by sound is:

$$ \text{Total distance} = \text{AB} + \text{DB} = 900 + (900 - 5v) = 1800 - 5v \text{ meters}. $$

But we know the total distance is 1650 meters. Set them equal:

$$ 1800 - 5v = 1650 $$

Solve for $$ v $$:

Subtract 1800 from both sides:

$$ -5v = 1650 - 1800 $$

$$ -5v = -150 $$

Divide both sides by -5:

$$ v = \frac{-150}{-5} = 30 \text{ m/s}. $$

Hence, the speed of the engine is 30 m/s.

So, the answer is Option D.