A boat moving at 15 km/h relative to water is crossing a river 1.5 km wide. The river flows at 5.0 km/h

a) if the boat heads directly for the opposite shore, how long will the trip take?
b)Where will the boat arrive on the opposite shore?
c)If the boat is to reach a point directly across from the starting point , in which direction should the boat head?
d)How long will the crossing take in case c)?

To solve these problems, we need to break down the boat's motion relative to the ground into its horizontal and vertical components. Let's go through each question step by step:

a) To find the time taken to cross the river, we need to determine the time taken to cover the horizontal distance (1.5 km) while accounting for the boat's velocity relative to the water (15 km/h) and the river's flow (5.0 km/h).

First, let's calculate the magnitude of the boat's horizontal velocity (Vh). We know that the magnitude of the velocity of the boat relative to the water is 15 km/h.

Vh = 15 km/h

Next, let's calculate the magnitude of the boat's vertical velocity (Vv). We know that the river's flow velocity is 5.0 km/h.

Vv = 5.0 km/h

Now, since the boat travels at an angle (θ) to the direction of the river's flow, we can use trigonometry to determine the magnitude of the boat's resultant velocity (V).

V = √(Vh² + Vv²)

V = √((15 km/h)² + (5.0 km/h)²)

V ≈ √(225 km²/h² + 25.0 km²/h²)

V ≈ √(250 km²/h²)

V ≈ 15.81 km/h

Now that we have the magnitude of the boat's resultant velocity, we can use it to find the time taken to cross the river.

Time = Distance / Velocity

Time = 1.5 km / 15.81 km/h ≈ 0.095 hours ≈ 5.7 minutes

So, it will take approximately 0.095 hours or 5.7 minutes for the boat to cross the river if it heads directly for the opposite shore.

b) To determine where the boat arrives on the opposite shore, we need to calculate the boat's actual displacement relative to the ground.

Displacement = Vh * Time

Displacement = 15 km/h * 0.095 hours ≈ 1.425 km

Since the boat's actual displacement is slightly less than the width of the river (1.5 km), it will arrive slightly downstream from the opposite shore.

c) If the boat is to reach a point directly across from the starting point, it needs to take a path that cancels out the effect of the river's flow. To do this, the boat should head slightly upstream from a straight line between the starting and ending points.

d) If the boat heads slightly upstream from a straight line, the distance it needs to travel will be longer. However, since the boat's horizontal velocity remains the same (15 km/h), the time taken to cross the river will still be the same. Therefore, it will still take approximately 0.095 hours or 5.7 minutes to cross the river in this case.

JohnX.

Please use my prior response as a starting point. We are not going to do the work for you.

The first line of the previous response tells you the answer to a.
The secondline tells you the answer to b.
The last part tells you how to get c and d.
I will be happy to critique your work or thinking