A motorboat heads due east at 8.5 m/s across a river that flows to the south at a speed of 2.5 m/s.

a. What is the velocity of the boat relative to a stationary observer on the shore?
B. If the river is 1360 m wide, how long does it take the boat to cross the river?
C. How far downstream will the boat have traveled when it reaches the other side?

The 8.5 m/s east motorboat velocity is actually with respect to the water. It may be pointed or headed east, but that is not the direction it goes. It drifts downstream with the water.

(a) The velocity with respect to shore is the vector sum of 8.5 m/s east and 2.5 m/s south. The path makes an angle of arctan 2.5/8.5 = 16.4 degrees away from the straight-across direction.

(b) 1360 m/8.5 m/s = 160 s

(c) 2.5 m/s * 160 s = 400 m

a. To determine the velocity of the boat relative to a stationary observer on the shore, we need to use vector addition.

Let's assume the eastward direction is positive and northward direction is negative.

The velocity of the boat with respect to the stationary observer can be found by adding the velocity of the boat relative to the water (8.5 m/s east) and the velocity of the water (2.5 m/s south).

Using vector addition, the resulting velocity can be found using the Pythagorean theorem:

v = √(v₁² + v₂²)

v = √((8.5 m/s)² + (-2.5 m/s)²)

v = √(72.25 m²/s² + 6.25 m²/s²)

v = √78.5 m²/s²

v ≈ 8.86 m/s

Therefore, the velocity of the boat relative to a stationary observer on the shore is approximately 8.86 m/s.

b. To determine how long it takes the boat to cross the river, we can use the concept of relative velocity.

The boat needs to cover the width of the river while drifting downstream due to the current. The time taken can be calculated using the formula:

Time = Distance / Relative Velocity

The distance is given as 1360 m.

The relative velocity of the boat with respect to the shore observer is 8.86 m/s (as calculated in part a).

Time = 1360 m / 8.86 m/s

Time ≈ 153.54 seconds

Therefore, it will take the boat approximately 153.54 seconds to cross the river.

c. To determine how far downstream the boat will have traveled when it reaches the other side, we can use the time it takes to cross the river and the downstream velocity of the river.

The downstream distance can be calculated by multiplying the time taken to cross the river (153.54 seconds) by the downstream velocity (2.5 m/s).

Distance downstream = Time × Downstream velocity

Distance downstream = 153.54 s × 2.5 m/s

Distance downstream ≈ 383.85 meters

Therefore, the boat will have traveled approximately 383.85 meters downstream when it reaches the other side of the river.

a. To find the velocity of the boat relative to a stationary observer on the shore, we can use vector addition. The boat's velocity can be split into two components: one in the east direction (due to its own motion) and one in the south direction (due to the river's flow).

Using the Pythagorean theorem, we can find the magnitude of the resulting velocity. Let's call the velocity of the boat relative to the observer V.

V^2 = V_east^2 + V_south^2

where V_east is the velocity of the boat in the east direction (8.5 m/s) and V_south is the velocity of the river's flow in the south direction (2.5 m/s).

V = sqrt(8.5^2 + 2.5^2)

V ≈ 8.87 m/s

Therefore, the velocity of the boat relative to a stationary observer on the shore is approximately 8.87 m/s.

b. To find how long it takes for the boat to cross the river, we can use the equation:

Time = Distance / Velocity

The distance is given as 1360 m, and the boat's velocity in the east direction is 8.5 m/s. Since the boat is moving diagonally, it will take longer to cross the river than if it were moving straight across.

Time = 1360 m / 8.5 m/s

Time ≈ 160 seconds

Therefore, it will take approximately 160 seconds for the boat to cross the river.

c. To find how far downstream the boat will have traveled when it reaches the other side, we need to find the southward displacement of the boat during the crossing. This can be calculated using the time it takes to cross the river (which we found in part b) and the velocity of the river's flow, which is given as 2.5 m/s.

Displacement = Velocity_south * Time

Displacement = 2.5 m/s * 160 seconds

Displacement = 400 m

Therefore, the boat will have traveled approximately 400 m downstream when it reaches the other side.

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