Question

Two submarines start from the same spot and travel in opposite directions, one at \(115 \mathrm{km} /\) day and the other at \(182 \mathrm{km} /\) day. How long will it take for the submarines to be \(1470 \mathrm{km}\) apart?

Short answer

It will take the submarines approximately 4.95 days to be 1470 km apart.

Step-by-step solution

Determine the Combined Rate of Separation

Since the submarines are traveling in opposite directions, their rates add up. The combined rate of separation is the sum of the individual rates of the two submarines, which is calculated by adding their speeds together. The first submarine travels at a rate of 115 km/day and the second at a rate of 182 km/day. Therefore, the combined rate of separation is 115 km/day + 182 km/day.

Calculate the Combined Rate of Separation

Add the rates from Step 1 to find the total rate at which the submarines are separating. That is, 115 km/day + 182 km/day = 297 km/day.

Determine the Time to Reach the Desired Separation

To find out how long it will take for the submarines to be 1470 km apart, we divide the total distance by the combined rate of separation. The formula for time is time = distance / rate. We use the distance of 1470 km and the combined rate of 297 km/day.

Calculate the Time

Divide the total separation distance by the combined rate of separation to find the time: time = 1470 km / 297 km/day.

Simplify the Calculation

Perform the division to calculate the time it takes for the submarines to be 1470 km apart: 1470 km / 297 km/day which simplifies to 4.9495 days. Since the context of the problem does not explicitly require an approximation or rounding, we can express the time either in a rounded form or as an exact decimal depending on the requirement.

Key concepts

Combined Rate of Separation

When dealing with problems involving two entities moving away from each other, it's crucial to understand the concept of combined rate of separation. This is the speed at which the distance between the entities is increasing. In the scenario with two submarines traveling in opposite directions, their individual speeds contribute to the overall rate at which they separate.

To calculate this, simply add their speeds together. If submarine A travels at 115 km/day and submarine B at 182 km/day, their combined rate of separation is the sum of these two speeds, equating to 297 km/day. This rate helps us understand how quickly the gap between the submarines grows over time.

Relative Velocity

Relative velocity is a measure of the velocity of one object as observed from another moving object. In the context of our problem, if one submarine is observed from the reference point of the other, the apparent rate at which the first one moves away is the sum of their individual speeds because they are moving in opposite directions.

Thus, the relative velocity concept is closely allied with the combined rate of separation. When both objects move towards each other, their speeds get subtracted to compute relative velocity. However, since they're moving away from each other, we add the speeds, resulting in a relative velocity of 297 km/day. Understanding relative velocity is critical when analyzing the dynamics of moving objects in physics and engineering.

Distance-Time Relationship

The distance-time relationship is an essential principle that defines the linkage between the distance traveled, the speed (or rate) at which an object travels, and the time it takes to cover that distance. The relationship is grounded on the formula: \[ \text{time} = \frac{\text{distance}}{\text{rate}} \]

This formula tells us that the time taken to travel a certain distance is equal to that distance divided by the speed. In our submarine scenario, to find out how long it will take for them to be 1470 km apart, we divide the desired separation distance by the combined rate of separation. Applying the distance-time relationship, we get:\[ \text{time} = \frac{1470 \, \text{km}}{297 \, \text{km/day}} \]After calculating, we find that it takes approximately 4.9495 days for the submarines to reach the required separation. This demonstrates how distance, time, and speed/rate are interdependent, and how altering one variable can affect the others.