A child in danger of drowning in a river is being carried...

A child in danger of drowning in a river is being carried downstream by a current that has a speed of $2.50 \mathrm{~km} / \mathrm{h}$. The child is $0.600 \mathrm{~km}$ from shore and $0.800 \mathrm{~km}$ upstream of a boat landing when a rescue boat sets out. (a) If the boat proceeds at its maximum speed of $20.0 \mathrm{~km} / \mathrm{h}$ relative to the water, what heading relative to the shore should the pilot take? (b) What angle does the boat velocity make with the shore? (c) How long does it take the boat to reach the child?

A child in danger of drowning in a river is being carried downstream by a current that has a speed of $2.50 \mathrm{~km} / \mathrm{h}$. The child is $0.600 \mathrm{~km}$ from shore and $0.800 \mathrm{~km}$ upstream of a boat landing when a rescue boat sets out.
(a) If the boat proceeds at its maximum speed of $20.0 \mathrm{~km} / \mathrm{h}$ relative to the water, what heading relative to the shore should the pilot take? (b) What angle does the boat velocity make with the shore? (c) How long does it take the boat to reach the child?

Key Concepts

Time Determination in Kinematics

Time determination in kinematics involves calculating the duration of travel by using the distance to be covered and the effective velocity component towards the target. In problems involving moving media, it is essential to compute the resultant velocity in the direction of the desired motion, then divide the distance by this velocity to estimate the time required for the interception or destination arrival.

Relative Velocity

Relative velocity involves analyzing the motion of an object with respect to a moving medium. In this context, it means understanding how the boat's speed relative to the water combines with the river's current to determine the boat's overall velocity relative to the shore. This concept is pivotal since the rescue boat's effective path is the vector sum of its own propulsion and the influence of the river's current.

Vector Decomposition

Vector decomposition is the process of breaking down a velocity (or any vector quantity) into its horizontal and vertical components. By decomposing the boat's velocity and the river current into orthogonal directions (typically parallel and perpendicular to the shore), it becomes possible to apply trigonometric functions to determine the needed heading and to calculate the effective speed in the direction of the target.

Heading Adjustment (Course Correction)

Heading adjustment refers to the method of changing the direction in which the boat is pointed to compensate for the lateral effect of the current, ensuring that the boat intercepts the drifting target. This involves determining the angle relative to a fixed reference (such as the shore) by which the boat must aim upstream or downriver, effectively offsetting the influence of the current.

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