Two boats, A and B, travel with a velocity of 4.90 m/s across a river of width 72.0 m. The river flows with a velocity of 2.50 m/s. Boat A travels the shortest distance and boat B travels in the shortest time. If both start at the same time, how much time will they take to cross the river?

The shortest time is to orient the boat at 90° to the shore. However, the boat will be carried downstream on arriving the other side.

The time required will be the width of the river divided by the velocity of the boat in still water.

The shortest distance is also the only way by which the boat arrives directly opposite to the starting point.
The boat must be oriented towards upstream in such a way that the component parallel to the river cancels out the velocity of the river, or
4.9cos(θ)=2.5
where θ is the angle with the river.
Solving, &theta=59.3226° with the river.
The component of the boat's velocity perpendicular to the river is thus
4.9sin(θ) m/s
The time required is therefore
Width of river / (4.9sin(θ)) s.

To solve this problem, we can break it down into two components: the horizontal component (across the river) and the vertical component (along the river's flow).

For boat A:
- The velocity across the river is 4.90 m/s (same as its overall velocity).
- The velocity along the river is 2.50 m/s (same as the river's flow).

The time it takes for boat A to cross the river can be calculated using the equation:

time = distance / velocity

The distance across the river for boat A is equal to the width of the river, which is 72.0 m. Therefore:

time A = 72.0 m / 4.90 m/s

For boat B:
- The velocity across the river is the vector difference between the boat's velocity and the river's flow velocity.
- This can be calculated using vector subtraction:

velocity across the river B = sqrt((4.90 m/s)^2 - (2.50 m/s)^2)

The distance across the river for boat B is also equal to the width of the river, which is 72.0 m. Therefore:

time B = 72.0 m / (velocity across the river B)

Now we can calculate the times for both boats A and B.

time A = 72.0 m / 4.90 m/s = 14.69 seconds (rounded to two decimal places)

time B = 72.0 m / (velocity across the river B) = 72.0 m / sqrt((4.90 m/s)^2 - (2.50 m/s)^2) = 15.98 seconds (rounded to two decimal places)

Therefore, boat A will take approximately 14.69 seconds and boat B will take approximately 15.98 seconds to cross the river.

To find out how much time it will take for both boats to cross the river, we can use the concept of relative velocity.

Let's start with boat A. Boat A is traveling at a velocity of 4.90 m/s across the river. We can break down this velocity into two components: one along the river's direction and one perpendicular to it.

The component of boat A's velocity along the river's direction is the same as the river's velocity, which is 2.50 m/s. This is because the boat is moving directly across the river and is not affected by the river's flow in this direction.

The component of boat A's velocity perpendicular to the river's direction is the part that allows the boat to move towards the other bank of the river. Because the boat is moving in a right-angled triangle, we can use the Pythagorean theorem to find this component:

Velocity perpendicular to the river's direction = √(Velocity across river)^2 - (River's velocity)^2

Velocity perpendicular to the river's direction = √(4.90 m/s)^2 - (2.50 m/s)^2
= √24.01 m^2/s^2 - 6.25 m^2/s^2
= √17.76 m^2/s^2
≈ 4.216 m/s

Now, let's move on to boat B. Boat B is traveling in the shortest time, which means it will take a path that is directly across the river without factoring in the river's flow. Therefore, boat B will have a velocity equal to the velocity across the river:

Velocity of boat B = Velocity across river = 4.90 m/s

Now that we have the velocities for both boats, we can calculate the time it takes for each boat to cross the river using the formula:

Time = Distance / Velocity

For boat A:

Time for boat A to cross the river = Width of the river / Velocity perpendicular to the river's direction
= 72.0 m / 4.216 m/s
≈ 17.095 s

For boat B:

Time for boat B to cross the river = Width of the river / Velocity across the river
= 72.0 m / 4.90 m/s
≈ 14.694 s

Therefore, boat A will take approximately 17.095 seconds to cross the river, while boat B will take approximately 14.694 seconds.