A boat travels across a river. The boat's engine drives it at 4.0 m/s due north. The river current flows at 3.0 m/s due east. (a) Calculate the resultant sp...

(a) Resultant speed of the boat [2]

The boat's engine drives it at 4.0 m/s north and the current pushes it at 3.0 m/s east. These two velocity components are perpendicular, so the resultant is found using Pythagoras:

\( v = \sqrt{4.0^2 + 3.0^2} = \sqrt{16 + 9} = \sqrt{25} \) [1]

\( v = 5.0 \text{ m/s} \) [1]

This is a 3-4-5 right triangle. The resultant velocity is directed northeast at an angle to the north bank.

(b) Time to cross the river [2]

The river is 80 m wide (measured north-south). The boat's northward component is 4.0 m/s, which is the speed responsible for crossing:

\( t = \frac{\text{width}}{\text{speed across}} = \frac{80}{4.0} \) [1]

\( t = 20 \text{ s} \) [1]

The eastward current does not affect the time to cross, only where the boat lands.

(c) Distance carried downstream [2]

During the 20 s crossing, the current carries the boat eastward:

\( \text{downstream distance} = v_{\text{current}} \times t = 3.0 \times 20 \) [1]

\( = 60 \text{ m} \) [1]

(d) Why the boat does not travel straight north [2]

The river current exerts a sideways force on the boat, pushing it downstream (eastward). [1]

The actual path of the boat is determined by the vector sum of the boat's velocity (north) and the current's velocity (east). Since these point in different directions, the resultant velocity is at an angle, and the boat follows a diagonal path rather than going straight north. [1]