A swimmer, capable of swimming at a speed of 1.24 m/s in still water (i.e., the swimmer can swim with a speed of 1.24 m/s relative to the water), starts to swim directly across a 2.24-km-wide river. However, the current is 0.553 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?

a. Vsc = Vs + Vc = 1.24i + 0.553, Q1.

Tan A = Y/X = 1.24/0.553 = 2.240
A = 66o CCW.

V = Y/sin A = 1.24/sin66 = 1.36m/s

d1 = 2.24km/sin66 = 2.45 km = 2450 m. to
cross

d1 = V*t
t = d1/V = 2450/1.36 = 1802 s. = 30 Min.
to cross.

b. d2 = 2450*Cos66 = 997 m. Downstream.

To solve this problem, we need to use the concept of relative velocity. Relative velocity is the combined effect of the swimmer's velocity and the current's velocity.

(a) To calculate how long it takes for the swimmer to cross the river, we can use the formula:
time = distance / velocity

Since the swimmer is traveling directly across the river, the distance to be covered is the width of the river, which is given as 2.24 km. However, we need to convert this distance to meters, as the swimmer's velocity is given in m/s.
1 km = 1000 m
Therefore, the distance across the river is 2.24 km * 1000 m/km = 2240 m.

Now, to find the effective velocity (the velocity with respect to the shore), we subtract the current velocity from the swimmer's velocity:
effective velocity = swimmer's velocity - current velocity
= 1.24 m/s - 0.553 m/s
= 0.687 m/s

Using the formula, time = distance / velocity, we can calculate the time it takes for the swimmer to cross the river:
time = 2240 m / 0.687 m/s
time ≈ 3259.6 seconds

Therefore, it takes approximately 3259.6 seconds for the swimmer to cross the river.

(b) To calculate how far downstream the swimmer will be upon reaching the other side of the river, we can use the formula:
distance downstream = current velocity * time

Substituting the given values, we get:
distance downstream = 0.553 m/s * 3259.6 seconds
distance downstream ≈ 1799.04 meters

Therefore, the swimmer will be approximately 1799.04 meters downstream when reaching the other side of the river.

To find the answers to these questions, we need to break down the swimmer's motion into horizontal and vertical components.

Let's start with part (a):

(a) How long does it take the swimmer to cross the river?

To determine the time it takes for the swimmer to cross the river, we need to find the swimmer's net velocity (velocity relative to the riverbank).

The swimmer's net velocity can be calculated using vector addition. The horizontal component of the swimmer's velocity is equal to the swimmer's speed in still water (1.24 m/s), and the vertical component is the speed of the current (0.553 m/s).

So, the swimmer's net velocity can be found using the Pythagorean theorem:

Net velocity = √(horizontal velocity^2 + vertical velocity^2)

Net velocity = √(1.24^2 + 0.553^2) m/s

Now we can calculate the net velocity:

Net velocity = √(1.5376 + 0.305809) m/s

Net velocity ≈ √1.843 m/s

Net velocity ≈ 1.358 m/s

Now, to find the time it takes for the swimmer to cross the river, we can divide the width of the river (2.24 km) by the net velocity:

Time = Distance / Velocity

Time = 2.24 km / 1.358 m/s

However, we need to convert the distance to meters to match the units of velocity:

2.24 km = 2.24 * 1000 m = 2240 m

Time = 2240 m / 1.358 m/s

Finally, we can calculate the time it takes the swimmer to cross the river:

Time ≈ 1650.07 seconds

Therefore, it takes the swimmer approximately 1650.07 seconds to cross the river.

Now let's move on to part (b):

(b) How far downstream will the swimmer be upon reaching the other side of the river?

To find out how far downstream the swimmer will be, we need to determine the displacement caused by the river's current while the swimmer crosses.

The downstream displacement can be found by multiplying the time it takes to cross the river by the velocity of the current:

Displacement = Time × Velocity

Displacement = 1650.07 seconds × 0.553 m/s

Displacement ≈ 912.47 meters

Therefore, the swimmer will be approximately 912.47 meters downstream upon reaching the other side of the river.