A kayaker needs to paddle north across a 100-m- wide harbor. The tide is going out, creating a t

step 1 So you have a kayaker trying to cross a harbor where the tide is going out at two meters per sec

A kayaker needs to paddle north across a 100-m- wide harbor....

A kayaker needs to paddle north across a $100-\mathrm{m}-$ wide harbor. The tide is going out, creating a tidal current flowing east at $2.0 \mathrm{m} / \mathrm{s}$. The kayaker can paddle with a speed of $3.0 \mathrm{m} / \mathrm{s}$ a. In which direction should he paddle in order to travel straight across the harbor? b. How long will it take him to cross?

A kayaker needs to paddle north across a $100-\mathrm{m}-$ wide harbor. The tide is going out, creating a tidal current flowing east at $2.0 \mathrm{m} / \mathrm{s}$. The kayaker can paddle with a speed of $3.0 \mathrm{m} / \mathrm{s}$
a. In which direction should he paddle in order to travel straight across the harbor?
b. How long will it take him to cross?

Key Concepts

Relative Velocity

Relative velocity refers to the velocity of an object as observed from a moving frame of reference. In this context, it involves combining the kayaker's paddling speed (relative to the water) with the tidal current’s velocity (relative to the ground) to find the actual path and speed of the kayak relative to the harbor.

Vector Addition

Vector addition is the process of combining two or more vectors to determine a resultant vector. For the kayaker, this means adding the vector representing his paddling direction and speed with the vector representing the tidal current to obtain his overall velocity across the harbor.

Decomposing Vectors into Components

Decomposing vectors into their perpendicular components allows one to analyse each directional influence separately. In this problem, the kayaker’s paddling vector is split into a northward component that contributes to crossing the harbor and an eastward component that must counteract the tidal current, ensuring a straight-line crossing.

Trigonometry in Vector Problems

Trigonometry is used to determine the necessary angle at which the kayaker must paddle so that his northward component cancels the effect of the eastward tidal current. By using sine, cosine, or tangent functions, one can relate the magnitudes of the vector components to the overall paddling speed.

Time Calculation Using Distance and Velocity

The time taken to cross the harbor is calculated using the relation between distance, speed, and time. Specifically, once the effective northward velocity is determined from the vector decomposition, the time can be computed by dividing the harbor’s width by this velocity component.

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