when a volcanoe erupts it emits smoke and lava in all directions with a velocity of approximately 280m/s. the height of the volcanoe is 5km and there is a house located at a distance of 8km from the centre of the mountain.5 s after the eruption a group of people start fleeing from the house in a car.that can run at a maximum speed of 90km/hr.assuming that they travel in a straight line away from the volcanoe, what additional distance should they have covered to escape the fireball?

To determine how far the group of people needs to travel to escape the fireball from the volcano, we need to analyze the situation step by step. First, we’ll calculate how far the volcanic eruption's fireball travels in the first 5 seconds before the people start fleeing. Then, we’ll find out how far the fireball will continue to travel while the people are escaping. Finally, we can calculate the additional distance they need to cover to ensure their safety.

Step 1: Calculate the distance traveled by the fireball in the first 5 seconds

The fireball from the volcano travels at a speed of 280 m/s. To find out how far it travels in 5 seconds, we can use the formula:

Distance = Speed × Time

Substituting the values:

Distance = 280 m/s × 5 s = 1400 m

This means that in the first 5 seconds after the eruption, the fireball travels 1400 meters, or 1.4 kilometers.

Step 2: Determine the total distance from the volcano to the house

The height of the volcano is 5 km, and the house is located 8 km away horizontally from the volcano. To find the direct distance from the volcano to the house, we can use the Pythagorean theorem:

Distance = √(Height² + Horizontal Distance²)

Substituting the values:

Distance = √(5² + 8²) = √(25 + 64) = √89 ≈ 9.43 km

Step 3: Calculate the remaining distance to the house after 5 seconds

After 5 seconds, the fireball has traveled 1.4 km towards the house, so we need to subtract this from the total distance:

Remaining distance = Total distance - Distance traveled by fireball

Remaining distance = 9.43 km - 1.4 km = 8.03 km

Step 4: Determine how far the car can travel in the time it takes to escape

The car can travel at a maximum speed of 90 km/h. First, we need to convert this speed into meters per second for consistency:

Speed in m/s = Speed in km/h × (1000 m / 1 km) × (1 h / 3600 s)

Speed in m/s = 90 × (1000 / 3600) ≈ 25 m/s

Now, we need to find out how long it takes for the car to travel the remaining distance of 8.03 km (or 8030 meters):

Time = Distance / Speed

Time = 8030 m / 25 m/s ≈ 321.2 seconds

Step 5: Calculate how far the fireball travels during this time

While the car is traveling, the fireball continues to move. We can calculate how far the fireball travels in 321.2 seconds:

Distance traveled by fireball = Speed × Time

Distance traveled by fireball = 280 m/s × 321.2 s ≈ 89,936 m or 89.94 km

Final Calculation: Determine the additional distance needed to escape

Since the fireball travels significantly farther than the distance to the house, we need to ensure that the car travels beyond the distance the fireball can cover. The fireball will reach the house long before the car can escape, so the group needs to cover more distance.

To escape safely, they should ideally be at least as far away as the fireball travels, which is approximately 89.94 km. Since they start at 8 km away, the additional distance they need to cover is:

Additional distance = Distance traveled by fireball - Initial distance from the volcano

Additional distance = 89.94 km - 8 km = 81.94 km

Thus, to escape the fireball safely, the group should aim to cover an additional distance of approximately 81.94 kilometers away from the volcano.