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 =....................km?
To solve this problem, we can use vector addition to find the resultant velocity of the boat.
Let's break down the velocities involved:
1. Velocity of the river current: 3.1 km/hr to the east. We'll denote this as v1.
2. Velocity of the boat: 24.8 km/hr (with respect to the water).
3. Relative velocity of the boat with respect to the current: This is the effective velocity of the boat, which we'll denote as v2.
Now, we can find v2 by considering the vector addition of v1 and v2:
v2 = √(v1^2 + v^2)
= √((3.1 km/hr)^2 + (24.8 km/hr)^2)
≈ √(9.61 + 615.04)
≈ √624.65
≈ 25 km/hr (approximately)
Since we want the boat to reach the child as fast as possible, we want the boat to travel directly towards the child. Therefore, the angle between the boat's velocity and the direction of the river current would be 90 degrees.
Using the Pythagorean theorem, we can find the distance the boat travels along the river before reaching the child:
Distance = (Speed of the boat relative to the current) × (time)
= v2 × (time)
Since we're dealing with distances and speeds, the time cancels out, and we can simplify to:
Distance = v2 × time
To find the time it takes the boat to reach the child, we need to consider the horizontal component of the boat's velocity, which is v2. The horizontal distance the boat travels is 2.5 km, so we can set up the equation:
Distance = v2 × time
2.5 km = 25 km/hr × time
Solving for time:
time = 2.5 km / 25 km/hr
= 0.1 hr (or 6 minutes)
Finally, we can find the distance from the dock where the boat reaches the child:
Distance = (Speed of the boat relative to the current) × (time)
= v2 × time
= 25 km/hr × 0.1 hr
= 2.5 km
Therefore, the boat reaches the child at a distance of 2.5 kilometers from the dock.
To find the distance from the dock to where the boat reaches the child, we can break down the motion of the boat and the child relative to the river.
First, let's consider the motion of the child. The child is being carried downstream by the river current with a velocity of 3.1 km/hr to the east. Since the child is 2.5 km upstream from the dock, the time it takes for the child to reach the dock can be found using the equation:
time = distance / velocity = 2.5 km / (-3.1 km/hr) = -0.806 hr
Note that the negative sign indicates that the child is moving against the current and the actual time would be positive.
Now, let's consider the motion of the boat. The boat has a speed of 24.8 km/hr relative to the water, but it needs to take into account the current of the river. To reach the child as fast as possible, the boat should take the most direct path, which will have both a horizontal (eastward) and vertical (upstream) component of motion.
To find the optimum angle, we can use the concept of vector addition. The resultant velocity of the boat should be the vector sum of its velocity relative to the water and the velocity of the river current. Since the boat is moving at an angle, we can break down the velocity of the boat into horizontal and vertical components.
Let's denote the angle of the boat's velocity relative to the horizontal as θ (theta). The horizontal component can be found using the equation:
horizontal component = velocity relative to the water * cos(θ)
The vertical component can be found using the equation:
vertical component = velocity relative to the water * sin(θ)
Since the horizontal component needs to cancel out the effect of the river current, we have:
horizontal component = 3.1 km/hr
Now, let's calculate the vertical component using the boat's speed relative to the water:
vertical component = 24.8 km/hr * sin(θ)
Using these components, we can find the time it takes for the boat to reach the child. The time can be given by:
time = distance / velocity = 0.6 km / (24.8 km/hr * sin(θ))
Now, the total time taken for the boat to reach the child is the sum of the time taken by the child and the time taken by the boat:
total time = child's time + boat's time
This can be represented as:
-0.806 hr + 0.6 km / (24.8 km/hr * sin(θ))
To minimize the total time, we need to find the value of θ that minimizes this expression.
Once we find the value of θ, we can substitute it back into the equation for the horizontal component to find the distance from the dock where the boat reaches the child:
horizontal component = 24.8 km/hr * cos(θ)
This will give us the distance from the dock to where the boat reaches the child.