A boat running downstream covers a distance of 16 km in 2 hours while for covering the same distance upstream, It takes 4 hours. What is the speed of the boat in still water? 

A boat running downstream covers a distance of 16 km in 2 hours while for covering the same distance upstream, It takes 4 hours. What is the speed of the boat in still water?

Let's denote the speed of the boat in still water as "b" km/h, and the speed of the stream as "s" km/h.

When the boat is going downstream, its effective speed is the sum of the speed of the boat and the speed of the stream, "b + s", and when it's going upstream, its effective speed is the difference, "b - s".

We're given that downstream, the boat covers 16 km in 2 hours, so:

b + s = 16/2 = 8

And upstream, it covers the same 16 km in 4 hours:

b - s = 16/4 = 4

Now we have a system of equations:

b + s = 8 b - s = 4

Adding the two equations:

(b + s) + (b - s) = 8 + 4 2b = 12

Dividing both sides by 2:

b = 6

So, the speed of the boat in still water is 6 km/h.

Relative Speed Problems in Mathematics

Relative speed problems in mathematics involve calculating the speed or velocity of one object relative to another. These types of problems commonly occur in physics, engineering, and everyday scenarios involving moving objects. Here's a general approach to solving relative speed problems:

1. Identify the objects involved: Determine which objects are moving and in what directions. For example, two cars traveling in opposite directions, a person walking towards a stationary object, etc.

2. Define the frame of reference: Decide which object or point of reference to use as a basis for calculating relative speeds. Often, one of the moving objects or a stationary point is chosen as the frame of reference.

3. Understand relative velocity: Relative velocity is the velocity of one object with respect to another. It is calculated by subtracting the velocity of the second object from the velocity of the first. If the objects are moving in the same direction, you subtract; if they're moving in opposite directions, you add.

4. Apply appropriate formulas: Depending on the scenario, you may use different formulas to calculate relative speed. For example:

   • If two objects are moving in the same direction, you subtract their speeds.
   • If two objects are moving in opposite directions, you add their speeds.
   • If one object is stationary, its speed is considered to be 0, and you only need to consider the speed of the moving object.
   • If one object is moving and the other is stationary, the relative speed is simply the speed of the moving object.

5. Convert units if necessary: Ensure that all speeds are in the same units before performing calculations.

6. Check your answer: Ensure that your answer makes sense in the context of the problem. For example, speeds cannot be negative unless they represent direction.

Here's an example problem to illustrate:

Example: Two trains A and B are traveling towards each other on parallel tracks. Train A is traveling at 60 km/h, and train B is traveling at 80 km/h. If they are initially 200 km apart, how long will it take for them to meet?

Solution:

1. Identify the objects involved: Train A and Train B.
2. Define the frame of reference: Let's consider the motion from the viewpoint of Train A or Train B. Since both trains are moving towards each other, it doesn't matter which one we choose.
3. Understand relative velocity: Since the trains are moving towards each other, we'll add their speeds to find the relative speed.
4. Apply appropriate formulas: Relative speed = Speed of A + Speed of B = 60 km/h + 80 km/h = 140 km/h. Time = Distance / Relative speed = 200 km / 140 km/h = 1.43 hours.
5. Convert units if necessary: None needed in this case.
6. Check your answer: The time is positive and reasonable given the scenario.

So, it will take approximately 1.43 hours for the two trains to meet.