A child is in danger of drowning in the Merimac river. The Merimac river has a current of 3.1 km/hr to the east. The child is 0.6 km from the shore and 2.5 km upstream from the dock. A rescue boat with speed 24.8 km/hr (with respect to the water) sets off from the dock at the optimum angle to reach the child as fast as possible. How far from the dock does the boat reach the child?
d =
To find the distance from the dock to where the boat reaches the child, we can break down the motion into horizontal and vertical components.
Let's first find the time it takes for the boat to reach the child. We'll focus on the horizontal motion since the current affects the horizontal displacement.
The child is 2.5 km upstream from the dock, meaning the boat needs to travel against the current. The effective speed of the boat in the horizontal direction is the boat's speed relative to the water minus the current speed:
effective speed = boat speed - current speed
= 24.8 km/hr - (-3.1 km/hr)
= 27.9 km/hr
Now, we can calculate the time it takes for the boat to reach the child. We'll use the formula time = distance / speed, where distance is the horizontal displacement (2.5 km upstream) and speed is the effective speed:
time = distance / speed
= 2.5 km / 27.9 km/hr
≈ 0.0896 hours
Next, let's find the vertical displacement of the child. We'll use the formula vertical displacement = current speed × time:
vertical displacement = current speed × time
= 3.1 km/hr × 0.0896 hours
≈ 0.278 km
Finally, we can find the distance from the dock to where the boat reaches the child by using the Pythagorean theorem, considering the horizontal displacement and the vertical displacement as the sides of a right triangle:
d = √(horizontal displacement^2 + vertical displacement^2)
= √((0.6 km)^2 + (0.278 km)^2)
≈ √(0.36 km^2 + 0.077184 km^2)
≈ √0.437184 km^2
≈ 0.661 km
Therefore, the boat reaches the child approximately 0.661 km from the dock.