Question

A boat which sails at 10km /hr in still water starts chasing from 10km behind another one which sails at 4km/hr in the upstream direction After how long wiil it catchup if the scream is flowing at 2km/hr? a.4 b.2.5h c. 2h d. 3.5h

Short answer

Answer: 1 hour

Step-by-step solution

Determine the Effective Speed of Each Boat

The speed of the boats is given relative to the still water. To find the effective speed of each boat, we need to consider the speed of the stream. For the boat going upstream, the effective speed will be the difference between its speed and the speed of the stream. Effective speed of upstream boat = Speed of the boat - Speed of the stream = 4 km/hr - 2 km/hr = 2 km/hr. For the boat chasing from behind, it is going downstream with the flow of the stream. Therefore, its effective speed is the sum of its speed and the speed of the stream. Effective speed of downstream boat = Speed of the boat + Speed of the stream = 10 km/hr + 2 km/hr = 12 km/hr.

Calculate the Relative Speed

Now that we know the effective speed of each boat, we can calculate their relative speed. The relative speed is simply the difference between the speeds of the two boats. Relative Speed = Effective speed of downstream boat - Effective speed of upstream boat = 12 km/hr - 2 km/hr = 10 km/hr. The relative speed represents the speed at which the chasing boat is closing the gap between the two boats.

Calculate the Time to Catch Up

Now, we'll use the relative speed and distance between the two boats to calculate the time it takes for the chasing boat to catch up to the second boat. To do this, we can use the formula for time: Time = Distance / Relative Speed = In this case, the distance is 10 km (given in the exercise) and the relative speed is 10 km/hr (calculated in step 2). Time = 10 km / 10 km/hr = 1 hour Thus, it will take 1 hour for the first boat to catch up to the second boat. This answer is not among the multiple choices (a, b, c, or d) provided, which means there could be a mistake in the options or the exercise is incomplete/incorrect.

Key concepts

Relative Speed

In problems involving two moving objects, relative speed plays a crucial role in determining how quickly one object catches up to or moves away from the other. Relative speed is how fast one object is moving in comparison to another.
Here, you calculate it by taking the difference between the speeds of the two objects.
For example, if one object is moving faster, then the relative speed will be positive, indicating that the faster object can catch the other. In the scenario where the chasing boat has an effective speed of 12 km/hr and the boat moving upstream has an effective speed of 2 km/hr, the relative speed is:

• Relative Speed = 12 km/hr - 2 km/hr = 10 km/hr

This essentially means the chasing boat is closing the gap between the two at 10 kilometers each hour. Understanding this concept is critical when solving such distance and time-related problems, especially when the objects are moving in the same direction, just like the boats in our problem.

Upstream and Downstream

When a boat is navigating through a river, the terms upstream and downstream are often used to describe the direction of travel relative to the flow of the water.

• Upstream: This direction is against the flow of the river. It results in a reduction of the boat's speed by the speed of the current. For example, if a boat moves upstream at 4 km/hr in still water and the stream is flowing at 2 km/hr, the effective speed upstream will be: 4 km/hr - 2 km/hr = 2 km/hr
• Downstream: This direction is with the flow of the river, which increases the boat's speed by the speed of the current. Using the given data where a boat sails at 10 km/hr in still water and the stream is 2 km/hr, the effective downstream speed would be: 10 km/hr + 2 km/hr = 12 km/hr

In each case, understanding whether a boat is moving upstream or downstream is vital for accurately calculating its actual speed. These principles help ensure correct calculations for scenarios like our chase problem.

Effective Speed Calculation

Effective speed is the actual speed at which a boat, or any object, moves. It takes into account not just the speed of the object itself, but also the influence of external factors like the speed of water currents or wind.
To calculate effective speed:

• For upstream travel (against the current): Subtract the speed of the current from the boat's speed in still water. For instance, with a boat speed of 4 km/hr and a stream speed of 2 km/hr, the effective speed would be: 4 km/hr - 2 km/hr = 2 km/hr
• For downstream travel (with the current): Add the speed of the current to the boat's speed in still water. As seen in our scenario, with a boat speed of 10 km/hr and a stream speed of 2 km/hr, the effective speed is: 10 km/hr + 2 km/hr = 12 km/hr

Accurate determination of effective speed is essential in solving problems related to chasing, where changes in speed directly affect the time and distance calculations needed to determine when one object will catch another.