Solving Swimmer Downstream and Upstream Problems

This problem involves understanding how the speed of the stream affects the speed of a swimmer moving with or against the current. We need to use the concepts of relative speed to find the distance between two points.

Understanding Relative Speed in Water

• When the swimmer moves downstream, the speed of the stream adds to the swimmer's speed in still water. This is because the stream is helping the swimmer move faster.
• When the swimmer moves upstream, the speed of the stream subtracts from the swimmer's speed in still water. This is because the stream is resisting the swimmer's movement.

Setting up the Equations

Let's define the variables:

• Let $s$ be the speed of the swimmer in still water (in km/h).
• Let $v$ be the speed of the stream (in km/h). We are given $v = 2$ km/h.
• Let $d$ be the distance between point A and point B (in km).

We know that Distance $=$ Speed $\times$ Time.

• Downstream: The swimmer travels from A to B in 4 hours.
  • Downstream speed = Speed of swimmer in still water $+$ Speed of stream
  • Downstream speed = $s + v = s + 2$ km/h
  • Distance $d = (s + 2) \times 4$
• Upstream: The swimmer travels from B to A (the same distance) in 5 hours.
  • Upstream speed = Speed of swimmer in still water $-$ Speed of stream
  • Upstream speed = $s - v = s - 2$ km/h
  • Distance $d = (s - 2) \times 5$

Since the distance $d$ is the same in both cases, we can set the two expressions for $d$ equal to each other:

$(s + 2) \times 4 = (s - 2) \times 5$

Solving for the Swimmer's Speed

Now, we solve this equation to find the value of $s$, the speed of the swimmer in still water:

$4s + 8 = 5s - 10$

To solve for $s$, we can rearrange the terms:

$8 + 10 = 5s - 4s$

$18 = s$

So, the speed of the swimmer in still water is 18 km/h.

Calculating the Distance

Now that we have the speed of the swimmer in still water ($s = 18$ km/h), we can use either the downstream or upstream equation for distance to find $d$.

Using the downstream equation:

$d = (s + 2) \times 4$

$d = (18 + 2) \times 4$

$d = 20 \times 4$

$d = 80$ km

Using the upstream equation (just to double-check):

$d = (s - 2) \times 5$

$d = (18 - 2) \times 5$

$d = 16 \times 5$

$d = 80$ km

Both calculations give the same distance, 80 km.

Summary of Steps

1. Define variables for swimmer speed, stream speed, and distance.
2. Write expressions for downstream speed (swimmer + stream) and upstream speed (swimmer - stream).
3. Use the formula Distance = Speed $\times$ Time to write equations for the distance in both the downstream and upstream cases.
4. Equate the two distance expressions since the distance is the same.
5. Solve the resulting equation to find the swimmer's speed in still water.
6. Substitute the swimmer's speed back into either the downstream or upstream distance equation to calculate the distance.

The distance between A and B is 80 km.

Revision Table: Swimmer Speed Concepts

Additional Information: Speed, Time, and Distance Problems

Problems involving relative speed in water (like swimmers or boats in streams) are common applications of the basic speed, time, and distance formula ($D = S \times T$).

Key points to remember:

• Always identify the speed of the object in still water and the speed of the stream.
• Remember that downstream speed is the sum of these two speeds, and upstream speed is the difference.
• The distance traveled is often the same for the downstream and upstream journeys in these types of problems, allowing you to equate the distance expressions.
• Be careful with units (km/h, meters/second, etc.) and ensure they are consistent throughout the calculation.

These problems often require setting up and solving linear equations, as demonstrated in the solution above.