Question

1. a speedboat with a speed of 20 m/s crosses a 120 m river with a current of 6 m/s.

a. what is the velocity of the speedboat as measured by a person standing next to the river?
b. what is the total displacement when it reaches the other side of the river?
c. if the speedboat doubles its speed how will the downstream displacement change?
d. if the river current doubles how will the time it takes the speedboat to cross the river change?

Explanation:

Step1: Find the resultant velocity (a)

The boat's velocity across the river $v_x = 20$ m/s and the river - current velocity $v_y=6$ m/s. The resultant velocity $v$ as measured by a person on the river - bank is given by the Pythagorean theorem $v=\sqrt{v_x^{2}+v_y^{2}}$.
$v=\sqrt{20^{2}+6^{2}}=\sqrt{400 + 36}=\sqrt{436}\approx20.9$ m/s.

Step2: Find the time to cross the river

The time $t$ taken to cross the river of width $d = 120$ m with a velocity $v_x=20$ m/s across the river is $t=\frac{d}{v_x}$.
$t=\frac{120}{20}=6$ s.

Step3: Find the downstream displacement (b)

The downstream displacement $x_y$ is given by $x_y = v_y\times t$. Since $v_y = 6$ m/s and $t = 6$ s, $x_y=6\times6 = 36$ m. The total displacement $s$ when it reaches the other side of the river is $s=\sqrt{d^{2}+x_y^{2}}=\sqrt{120^{2}+36^{2}}=\sqrt{14400+1296}=\sqrt{15696}\approx125.29$ m.

Step4: Analyze the change in downstream displacement when speed doubles (c)

The time to cross the river $t=\frac{d}{v_x}$. If the speed of the boat $v_x$ doubles, the new time $t_{new}=\frac{d}{2v_x}=\frac{t}{2}$. The downstream displacement $x_y=v_y\times t$. With $v_y$ constant, when $t$ is halved, the new downstream displacement $x_{y - new}=v_y\times t_{new}=\frac{1}{2}v_y\times t$. So the downstream displacement is halved.

Step5: Analyze the change in crossing - time when river current doubles (d)

The time to cross the river $t=\frac{d}{v_x}$. The river - current velocity $v_y$ has no effect on the time taken to cross the river as long as the boat is moving straight across the river. So if the river current doubles, the time it takes the speedboat to cross the river remains the same.

Answer:

a. Approximately $20.9$ m/s
b. Approximately $125.29$ m
c. The downstream displacement is halved.
d. The time remains the same.