A boat propelled so as to travel with a speed of 0.50m/s in still water, moves directly (in a straight line) across a river that is 60m wide. The river flows with a speed of 0.30m/s. How long in seconds does it take the boat to cross the river?

Presumably the boat is aimed perpendicular to the parallel banks, but the current carries it downstream.

Just for orientation, let's say the river flows east, and the boat crosses to the north.

In that case, we can draw a diagram involving the relative velocities. The boat's speed is .5 at 90 deg.

The river's speed is .3 at 0 deg

So, the boat's resultant speed is sqrt(.5^2 + .3^2) = .583 at 59.1 deg

At that direction, the distance travelled is 60 sec 30.9 deg = 70m

70m/.583s = 2 minutes

To find the time it takes for the boat to cross the river, we need to consider the relative motion of the boat with respect to the river.

Let's break this problem into two parts: the boat's motion perpendicular to the direction of the river flow (across the river) and the boat's motion parallel to the direction of the river flow (along the river).

First, let's calculate the time it takes for the boat to travel across the river. We can use the formula:

Time = Distance / Speed

The distance across the river is given as 60m, and the speed of the boat is 0.50m/s. So, the time it takes for the boat to cross the river is:

Time across the river = 60m / 0.50m/s = 120 seconds

Next, let's calculate the time it takes for the boat to travel along the river. Since the boat's speed is the same as the river's flow speed (0.30m/s), there is no relative motion, and the time to travel along the river will be 60m / 0.30m/s = 200 seconds.

Now, let's calculate the total time to cross the river by adding the time across the river and the time along the river:

Total time to cross the river = Time across the river + Time along the river
Total time to cross the river = 120 seconds + 200 seconds = 320 seconds

Therefore, it takes the boat 320 seconds to cross the river.

To find the time it takes for the boat to cross the river, we can use the concept of relative velocity.

Step 1: Determine the horizontal component of the boat's velocity.
The boat is moving directly across the river, perpendicular to the direction of the river's flow. Therefore, the horizontal component of the boat's velocity remains unchanged at 0.50 m/s.

Step 2: Determine the effective velocity of the boat.
The effective velocity of the boat is the vector sum of its velocity relative to the still water and the river's velocity. Since the boat is moving across the river, the effective velocity will form a right triangle with the boat's velocity and the river's velocity as its legs.

Using the Pythagorean theorem, we can calculate the effective velocity:
effective_velocity = √(boat_velocity^2 + river_velocity^2)
effective_velocity = √(0.50^2 + 0.30^2)
effective_velocity = √(0.25 + 0.09)
effective_velocity = √0.34
effective_velocity ≈ 0.58 m/s

Step 3: Calculate the time it takes to cross the river.
To cross the 60 m wide river, we can use the formula: time = distance / velocity.

time = 60 m / 0.58 m/s
time ≈ 103.45 seconds

Therefore, it takes approximately 103.45 seconds for the boat to cross the river.