Solving Boat Speed Problems: Finding Speed in Still Water

This problem involves the concept of speed relative to a moving medium, specifically water. When a boat travels in a river, its speed is affected by the speed of the stream. We consider two cases: travelling along the stream (downstream) and travelling against the stream (upstream).

Understanding Downstream and Upstream Speed

• Downstream Speed: This is the speed of the boat relative to the riverbank when it moves in the same direction as the stream. It is the sum of the boat's speed in still water and the speed of the stream.
• Upstream Speed: This is the speed of the boat relative to the riverbank when it moves in the opposite direction to the stream. It is the difference between the boat's speed in still water and the speed of the stream.

Setting Up the Problem

Let:

• \( B \) = Speed of the boat in still water (in km/hr)
• \( S \) = Speed of the stream (in km/hr)

Then:

• Downstream speed = \( B + S \)
• Upstream speed = \( B - S \)

We use the fundamental relationship between speed, distance, and time:

\( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \)

Calculating Downstream Speed

The boat travels 15 km in 15 minutes along the stream (downstream).

• Distance = 15 km
• Time = 15 minutes

First, convert the time from minutes to hours:

\( \text{Time in hours} = \frac{15 \text{ minutes}}{60 \text{ minutes/hour}} = \frac{1}{4} \text{ hours} \)

Now, calculate the downstream speed:

\( \text{Downstream Speed} = \frac{15 \text{ km}}{1/4 \text{ hours}} = 15 \times 4 \text{ km/hr} = 60 \text{ km/hr} \)

So, we have our first equation:

\( B + S = 60 \quad \text{(Equation 1)} \)

Calculating Upstream Speed

The boat travels 10 km in 1 hour against the stream (upstream).

• Distance = 10 km
• Time = 1 hour

Calculate the upstream speed:

\( \text{Upstream Speed} = \frac{10 \text{ km}}{1 \text{ hour}} = 10 \text{ km/hr} \)

So, we have our second equation:

\( B - S = 10 \quad \text{(Equation 2)} \)

Solving for the Speed of the Boat in Still Water

We now have a system of two linear equations with two variables, \( B \) and \( S \):

1. \( B + S = 60 \)
2. \( B - S = 10 \)

To find \( B \), we can add Equation 1 and Equation 2:

\( (B + S) + (B - S) = 60 + 10 \)

\( B + S + B - S = 70 \)

\( 2B = 70 \)

Now, solve for \( B \):

\( B = \frac{70}{2} \)

\( B = 35 \text{ km/hr} \)

The speed of the boat in still water is 35 km/hr.

We can also find the speed of the stream \( S \) by substituting the value of \( B \) into either equation. Using Equation 1:

\( 35 + S = 60 \)

\( S = 60 - 35 \)

\( S = 25 \text{ km/hr} \)

This means the speed of the stream is 25 km/hr.

Summary of Results

The speed of the boat in still water is 35 km/hr.

Revision Table: Boat and Stream Concepts

Additional Information: Boat Speed Formulas

From the downstream and upstream speeds, we can derive direct formulas for the speed of the boat in still water and the speed of the stream.

Let:

• \( V_d \) = Downstream Speed
• \( V_u \) = Upstream Speed

We know that:

• \( V_d = B + S \)
• \( V_u = B - S \)

Adding these two equations:

\( V_d + V_u = (B + S) + (B - S) \)

\( V_d + V_u = 2B \)

So, the speed of the boat in still water is:

\( B = \frac{V_d + V_u}{2} \)

Subtracting the second equation from the first:

\( V_d - V_u = (B + S) - (B - S) \)

\( V_d - V_u = B + S - B + S \)

\( V_d - V_u = 2S \)

So, the speed of the stream is:

\( S = \frac{V_d - V_u}{2} \)

Using these formulas with the speeds calculated earlier (\( V_d = 60 \) km/hr, \( V_u = 10 \) km/hr):

\( B = \frac{60 + 10}{2} = \frac{70}{2} = 35 \text{ km/hr} \)

\( S = \frac{60 - 10}{2} = \frac{50}{2} = 25 \text{ km/hr} \)

These formulas provide a quick way to find the boat's speed in still water and the stream's speed once the downstream and upstream speeds are known.