A river flows toward the East at 1.5 m/s. Steve can swim 2.2 m/s in still water. If he wishes to cross the river so that his direction is northward, what direction relative to the water must he swim?

X = 1.5 m/s = Velocity of water.

Y = 2.2 m/s = Velocity of the swimmer.

Tan A = Y/X = 2.2/1.5 = 1.46667.
A = 55.7o N. of E.

Direction = 55.7o N. of W. = 34.3o W. of
N.

Numerixal solve

How long it will take him to cross the 75 m wide river

63s

Well, Steve is going to have a swim-tastic adventure here! To cross the river and go northward, he needs to compensate for the river's current flowing eastward. So, he should swim at an angle opposite to the direction of the river's flow. In other words, if the river is going east, Steve should swim slightly westward. This way, his northward swimming and the river's current will cancel each other out, and he will end up crossing the river in a northern direction. Just remember, Steve, don't go swimming in circles like a confused fish!

To determine the direction Steve must swim relative to the water, we need to consider the effect of the river on his motion.

Let's break down the motion into two components: the component of Steve's velocity parallel to the river flow (eastward) and the component perpendicular to the river flow (northward).

We can start by finding the eastward component of Steve's velocity. Since the river flows toward the East at a speed of 1.5 m/s and Steve can swim at a speed of 2.2 m/s in still water, the eastward component of Steve's velocity when he is in the river can be found by subtracting the speed of the river from his swimming speed:

Eastward component = Steve's swimming speed - River speed
= 2.2 m/s - 1.5 m/s
= 0.7 m/s

Now, let's consider the northward component of Steve's velocity. Since he wants to cross the river in a northward direction, we want this component to be 0, meaning he swims directly across the river.

Since he is swimming perpendicular to the river flow, the northward and eastward components of Steve's velocity are perpendicular to each other. This means we can use Pythagoras' theorem to find the resultant velocity (the hypotenuse of a right triangle) and the angle he needs to swim at (the angle opposite the northward component).

Resultant velocity = √(northward component^2 + eastward component^2)
Resultant velocity = √(0^2 + 0.7^2)
Resultant velocity = √0.49
Resultant velocity ≈ 0.7 m/s

To find the angle, we will use trigonometry. The angle can be found by taking the inverse tangent (arctan or tan^(-1)) of the ratio of the northward component to the eastward component:

Angle = arctan(northward component / eastward component)
Angle = arctan(0 / 0.7)
Angle ≈ arctan(0)
Angle ≈ 0 degrees

Therefore, Steve needs to swim directly eastward (0 degrees relative to the water) to achieve a northward motion across the river.