A boat travels at 3.8 m/s and heads straight across a river 240m wide at an angle of 145'. The river flows at 1.6 m/s.
a. What is the boat's resultant speed with respect to the river bank?
b. How long does ti take the boat to cross the river?
c. How far downstream or upstream is the bot when it reaches the other side?
To solve this problem, we can break it down into different components and then use vector addition to find the resultant speed and direction of the boat.
a. To find the boat's resultant speed with respect to the river bank, we need to calculate the boat's velocity vector relative to the river bank. First, we can break down the boat's velocity into its horizontal and vertical components. The horizontal component represents the boat's speed across the river, and the vertical component represents the boat's speed along the river.
Given:
Boat's speed across the river (horizontal component) = 3.8 m/s
River's speed = 1.6 m/s
To find the boat's horizontal component, we can use trigonometry. Since the angle between the boat's path and the river is given as 145 degrees, we need to find the cosine of 145 degrees.
Cos(145°) = adjacent side / hypotenuse
Cos(145°) = horizontal component / 3.8 m/s
To get the boat's horizontal component:
horizontal component = 3.8 m/s * cos(145°)
Now, we also know that the river's speed is its vertical component. Therefore, the boat's vertical component is 1.6 m/s.
Using vector addition, the resultant speed with respect to the river bank is the magnitude of the combined horizontal and vertical components of the boat's velocity:
resultant speed = square root of (horizontal component^2 + vertical component^2)
b. To find how long it takes for the boat to cross the river, we need to find the time it takes for the boat to travel a distance of 240 meters. We can use the formula: time = distance / speed.
Given:
Distance to cross the river = 240 meters
Boat's speed across the river (horizontal component) = 3.8 m/s (from part a)
Therefore, the time it takes for the boat to cross the river is:
time = distance / speed
time = 240 meters / 3.8 m/s
c. Finally, to find how far downstream or upstream the boat is when it reaches the other side, we need to calculate the boat's displacement along the river.
Given:
Boat's speed along the river (vertical component) = 1.6 m/s
Time taken to cross the river = calculated in part b
To find the boat's displacement along the river, we can use the formula: displacement = speed * time.
If the boat's speed along the river is 1.6 m/s, and the time taken to cross the river is calculated in part b, we can find the displacement using the formula:
displacement = 1.6 m/s * time taken (from part b)
Now, to determine whether the boat is downstream or upstream, we need to consider the angle between the boat's path and the river. If the angle is less than 90 degrees, the displacement will be downstream. If the angle is greater than 90 degrees, the displacement will be upstream.
By following these calculations, you should be able to find the answers to all three questions.