Solving Boat and Stream Speed Problems

This question involves calculating the speed of the water current based on the time taken by a boatman to travel upstream and downstream distances. Let's break down the concepts and solve the problem step-by-step.

Understanding Boat and Current Speeds

When a boat travels in water, its effective speed is affected by the speed of the water current. We define the following terms:

• Speed of the boat in still water: The speed at which the boat can travel in water that is not moving.
• Speed of the water current: The speed at which the water is flowing.
• Upstream speed: The effective speed of the boat when it travels against the direction of the water current. The current opposes the boat's movement.
• Downstream speed: The effective speed of the boat when it travels in the same direction as the water current. The current helps the boat's movement.

The relationships are:

• Upstream speed = Speed of boat in still water − Speed of water current
• Downstream speed = Speed of boat in still water + Speed of water current

Also, we know the fundamental relationship between distance, speed, and time:

$\text{Speed} = \frac{\text{Distance}}{\text{Time}}$

Setting up the Problem Equations

Let:

• $B$ be the speed of the boat in still water (in km/h).
• $C$ be the speed of the water current (in km/h).

From the given information:

Case 1: Travelling against water current (Upstream)

• Distance covered = 24 km
• Time taken = 6 hours
• Upstream speed = Speed of boat in still water − Speed of water current = $B - C$

Using the formula $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$:

$B - C = \frac{24 \text{ km}}{6 \text{ hours}}$

$B - C = 4$ km/h (Equation 1)

Case 2: Travelling along with water current (Downstream)

• Distance covered = 36 km
• Time taken = 6 hours
• Downstream speed = Speed of boat in still water + Speed of water current = $B + C$

Using the formula $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$:

$B + C = \frac{36 \text{ km}}{6 \text{ hours}}$

$B + C = 6$ km/h (Equation 2)

Now we have a system of two linear equations with two variables, $B$ and $C$:

1. $B - C = 4$
2. $B + C = 6$

Solving for the Speed of Water Current

We want to find the speed of the water current, which is $C$. We can solve this system of equations using elimination or substitution. The elimination method is straightforward here.

Add Equation 1 and Equation 2:

$(B - C) + (B + C) = 4 + 6$

$B - C + B + C = 10$

$2B = 10$

$B = \frac{10}{2}$

$B = 5$ km/h

So, the speed of the boat in still water is 5 km/h.

Now, substitute the value of $B$ (5 km/h) into either Equation 1 or Equation 2 to find $C$. Let's use Equation 2:

$B + C = 6$

$5 + C = 6$

$C = 6 - 5$

$C = 1$ km/h

Thus, the speed of the water current is 1 km/h.

Summary of Calculations

Solving $B - C = 4$ and $B + C = 6$ gives $C = 1$ km/h.

The speed of the water current is 1 km/h.

Revision Table: Key Concepts in Boat and Stream Problems

Additional Information on Relative Speed

Boat and stream problems are applications of the concept of relative speed. Relative speed is the speed of an object with respect to another object or a frame of reference.

• When the boat moves upstream, the current opposes its motion. The effective speed (upstream speed) is the difference between the boat's speed in still water and the current's speed. This is like two objects moving towards each other, but relative to the water.
• When the boat moves downstream, the current supports its motion. The effective speed (downstream speed) is the sum of the boat's speed in still water and the current's speed. This is like two objects moving in the same direction, but the speed adds up relative to a stationary point on the bank.

From the derived equations $B - C = \text{Upstream Speed}$ and $B + C = \text{Downstream Speed}$, we can also find the speeds directly:

• Speed of boat in still water ($B$) = $\frac{\text{Downstream Speed} + \text{Upstream Speed}}{2}$
• Speed of current ($C$) = $\frac{\text{Downstream Speed} - \text{Upstream Speed}}{2}$

In this problem, Upstream Speed = 4 km/h and Downstream Speed = 6 km/h.

• $B = \frac{6 + 4}{2} = \frac{10}{2} = 5$ km/h
• $C = \frac{6 - 4}{2} = \frac{2}{2} = 1$ km/h

This confirms our result for the speed of the water current.