To find the resultant velocity of the boat, we can use vector addition. Let's assume the boat is rowed along the x-axis, and the river's flow is along the y-axis.
The velocity of the boat relative to the ground (resultant velocity) can be found using the Pythagorean theorem:
Resultant velocity = √(velocity of boat squared + velocity of river squared)
The velocity of the boat is given as 8 km/hr, and the velocity of the river is given as 6 km/hr. Converting both to meters per second (m/s) for consistency:
Velocity of boat = 8 km/hr = (8 * 1000) m / (60 * 60) s ≈ 2.22 m/s Velocity of river = 6 km/hr = (6 * 1000) m / (60 * 60) s ≈ 1.67 m/s
Substituting these values into the formula:
Resultant velocity = √(2.22^2 + 1.67^2) m/s ≈ √(4.93 + 2.79) m/s ≈ √7.72 m/s ≈ 2.78 m/s (rounded to two decimal places)
Therefore, the resultant velocity of the boat is approximately 2.78 m/s.
To find how far downstream the boat will reach the opposite bank, we can use the concept of relative velocity. The boat's velocity relative to the ground is the resultant velocity we calculated (2.78 m/s), and the breadth of the river is given as 100 m.
The distance downstream the boat will reach can be calculated using the formula:
Distance = Velocity * Time
We need to find the time it takes for the boat to cross the river. Since the boat is moving perpendicular to the river's flow, the time can be determined by the ratio of the breadth of the river to the velocity of the river.
Time = Distance (across the river) / Velocity of river
Time = 100 m / 1.67 m/s ≈ 59.88 s (rounded to two decimal places)
Now we can calculate the distance downstream:
Distance downstream = Velocity * Time
Distance downstream = 2.78 m/s * 59.88 s ≈ 166.45 m (rounded to two decimal places)
Therefore, the boat will reach approximately 166.45 meters downstream from its starting point on the opposite bank of the river.