A swimmer, capable of swimming at a speed of 1.18 m/s in still water (i.e., the swimmer can swim with a speed of 1.18 m/s relative to the water), starts to swim directly across a 2.16-km-wide river. However, the current is 1.19 m/s, and it carries the swimmer downstream. (a) How long does it take the swimmer to cross the river? (b) How far downstream will the swimmer be upon reaching the other side of the river?
X = Vc = 1.19 m/s.
Y = Vs = 1.18 m/s
Vr = sqrt(Vc^2 + Vs^2) = 1.68 m/s =
Resultant velocity.
Tan A = 1.18/1.19 = 0.99160
A = 44.8o
a. d1=2.16/sin44.8 = 3.07 km. = 3070 m.
Vr*T = 3070.
1.68*T = 3070.
T = 1827 s. = 30.5 Min.
b. d2 = 2.16/Tan44.8 = 2.175km = 2175 m.
= Distance downstream.
To answer these questions, we need to understand the concept of relative velocity and the Pythagorean theorem. Let's break it down step by step:
(a) To find the time it takes for the swimmer to cross the river, we need to determine the effective velocity of the swimmer relative to the ground. This can be calculated as the vector sum of the swimmer's velocity relative to the water and the velocity of the current.
Using the Pythagorean theorem:
velocity of swimmer relative to the ground = √((velocity of swimmer relative to water)^2 + (velocity of current)^2)
Plugging in the given values:
velocity of swimmer relative to the ground = √((1.18 m/s)^2 + (1.19 m/s)^2)
Now we can calculate the time it takes using the formula:
time = distance / velocity
distance = 2.16 km = 2160 m
time = 2160 m / (velocity of swimmer relative to the ground)
(b) To determine how far downstream the swimmer will be upon reaching the other side of the river, we need to multiply the time calculated in part (a) by the velocity of the current. This will give us the distance the current carries the swimmer downstream.
distance downstream = time * velocity of current
Let's solve these equations step by step:
(a)
1. Calculate the velocity of the swimmer relative to the ground:
velocity of swimmer relative to the ground = √((1.18 m/s)^2 + (1.19 m/s)^2)
velocity of swimmer relative to the ground ≈ 1.68 m/s
2. Calculate the time it takes for the swimmer to cross the river:
time = 2160 m / (velocity of swimmer relative to the ground)
time ≈ 1286.3 seconds
Therefore, it takes approximately 1286.3 seconds for the swimmer to cross the river.
(b)
1. Calculate the distance downstream:
distance downstream = time * velocity of current
distance downstream = 1286.3 seconds * 1.19 m/s
distance downstream ≈ 1531.5 meters
Therefore, the swimmer will be approximately 1531.5 meters downstream when reaching the other side of the river.