Question

1. the riverboat crosses an 80m wide river with a boat velocity of 5 m/s east and a river current of 4 m/s south.

a. what is the velocity of the riverboat as measured by a person standing next to the river?
b. how long does it take the riverboat to cross the river?
c. how far down river will the boat move while it crosses?

Explanation:

Step1: Find resultant velocity for part a

Use Pythagorean theorem for perpendicular velocities. Let $v_x = 5$ m/s (east - horizontal) and $v_y=4$ m/s (south - vertical). The resultant velocity $v$ is given by $v=\sqrt{v_x^{2}+v_y^{2}}$.
$v=\sqrt{5^{2}+4^{2}}=\sqrt{25 + 16}=\sqrt{41}\approx 6.4$ m/s

Step2: Find time to cross river for part b

The width of the river $d = 80$m and the component of velocity that takes the boat across the river is $v_x=5$m/s. Using the formula $t=\frac{d}{v}$, where $d$ is the distance and $v$ is the relevant velocity component. So $t=\frac{80}{5}=16$s.

Step3: Find downstream distance for part c

The downstream velocity is $v_y = 4$m/s and the time to cross the river $t = 16$s. Using the formula $x=v_y\times t$. So $x=4\times16 = 64$m

Answer:

a. Approximately 6.4 m/s
b. 16 s
c. 64 m