Question

1. a waterfall is located 45.0 m downstream from where the boat is launched. the current speed is 3 m/s. what minimum boat speed is required to cross the 100 - m wide river before falling over the falls? show your calculations; use the simulation to check your prediction.
2. repeat the above calculations to determine the boat speed required to cross the 100 - m wide river in time if the current speed is 6 m/s and the waterfall was located 45.0 m downstream. again, check your predictions using the simulation.
3. a 40 m wide river has a current of 5 m/s. the maximum speed that a boat has is also 5 m/s. can this boat cross the river in such a way that it does not float downstream? why or why not?
4. challenge: river crossing problem level 1 (thephysicsaviary.com)

Explanation:

Step1: Analyze time - motion relationship

The time available to cross the river is determined by the downstream motion before reaching the waterfall. For the first case, the downstream distance $d = 45.0$ m and the current speed $v_{current}=3$ m/s. Using the formula $t=\frac{d}{v_{current}}$, we get $t=\frac{45}{3}=15$ s.

Step2: Calculate minimum boat speed

The width of the river $w = 100$ m. To cross the river in the available time $t$, we use the formula $v_{boat}=\frac{w}{t}$. Substituting $w = 100$ m and $t = 15$ s, we have $v_{boat}=\frac{100}{15}=\frac{20}{3}\approx6.67$ m/s.

Step3: For the second - case analysis

When $v_{current}=6$ m/s and $d = 45.0$ m, the time available $t=\frac{45}{6}=7.5$ s. The width of the river is still $w = 100$ m. Then $v_{boat}=\frac{100}{7.5}=\frac{40}{3}\approx13.33$ m/s.

Step4: Analyze the third - case

For a 40 - m wide river with $v_{current}=5$ m/s and $v_{boat}=5$ m/s. The boat can cross the river without floating downstream if it heads at an appropriate angle. The component of the boat's velocity perpendicular to the current and the component along the current are considered. The maximum component of the boat's velocity perpendicular to the current is when the boat heads at an angle such that the velocity along the current cancels out the current velocity. In this case, the boat can cross the river without being carried downstream by heading at an angle $\theta$ where $\sin\theta=\frac{v_{current}}{v_{boat}}$. Since $v_{current}=v_{boat} = 5$ m/s, $\theta = 90^{\circ}$ relative to the direction of the current is not possible. But by heading at an angle $\theta=\arcsin(1) = 90^{\circ}$ in a vector - addition sense (component - wise), the boat can cross without being carried downstream.

Answer:

1. Minimum boat speed in the first case: $\frac{20}{3}\approx6.67$ m/s
2. Minimum boat speed in the second case: $\frac{40}{3}\approx13.33$ m/s
3. The boat can cross the river without floating downstream. It needs to head at an appropriate angle such that the component of its velocity along the direction of the current cancels out the current velocity. Since $v_{current}=v_{boat}=5$ m/s, it is possible by adjusting the direction of the boat's motion.