Question

A kayaker needs to paddle north across a 100-m-wide harbor. The tide is going out, creating a tidal current that flows to the east at 2.0 m/s. The kayaker can paddle with a speed of 3.0 m/s. How long will it take him to cross?

Accepted Answer

Step 1: Recognize that the kayaker's motion is influenced by two components: the northward paddling velocity and the eastward tidal current. The kayaker's paddling speed is 3.0 m/s, and the tidal current flows east at 2.0 m/s. To determine the time to cross, focus only on the northward motion since the width of the harbor is given as 100 m. Step 2: The kayaker's northward velocity is the component of his paddling speed directed perpendicular to the tidal current. Since the kayaker must paddle at an angle to counteract the eastward current, the northward velocity can be calculated using trigonometry. The northward velocity is given by $$ v_{north} = v_{kayaker} \cdot \cos(	heta) $$, where $$ 	heta  is $$the angle the kayaker paddles relative to the north direction. Step 3: To counteract the eastward current, the kayaker must paddle at an angle such that the eastward component of his paddling velocity cancels the tidal current. This condition is expressed as $$ v_{kayaker} \cdot \sin(	heta) = v_{current} . $$Solve for $$ \sin(	heta) $$ using $$ \sin(	heta) = \frac{v_{current}}{v_{kayaker}} . $$Substitute $$ v_{current} = 2.0 \, 	ext{m/s} $$ and $$ v_{kayaker} = 3.0 \, 	ext{m/s}  to $$find $$ \sin(	heta) . $$Step 4: Use the Pythagorean identity $$ \sin^2(	heta) + \cos^2(	heta) = 1  to $$calculate $$ \cos(	heta) . $$Substitute $$ \sin(	heta) $$ from Step 3 into the identity to find $$ \cos(	heta) . $$Then, calculate the northward velocity $$ v_{north} $$ using $$ v_{north} = v_{kayaker} \cdot \cos(	heta) . $$Step 5: Once $$ v_{north}  is $$determined, calculate the time to cross the harbor using the formula $$ t = \frac{d}{v_{north}} $$, where $$ d = 100 \, 	ext{m}  is $$the width of the harbor. Substitute $$ v_{north} $$ into the equation to find the time required for the kayaker to cross.

A kayaker needs to paddle north across a 100-m-wide harbor. The tide is going out, creating a tidal current that flows to the east at 2.0 m/s. The kayaker can paddle with a speed of 3.0 m/s. How long will it take him to cross?

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Relative Velocity

Vector Addition

Time Calculation