A boat must cross a 260-m-wide river and arrive at a point 110 upstream from where it starts. To do so, the pilot must head the boat at a 45 degree upstream angle. THe current has speed 2.3 m/s.
a) What is the speed of the boat in still water?
b) How much time does the journey take?
To solve the problem, we can use the concept of vectors. Let's consider the velocity of the boat in still water as V and the velocity of the river current as C.
a) To find the speed of the boat in still water, we first need to find the component of the boat's velocity in the upstream direction. This can be found using the equation:
V_upstream = V * cos(45°)
Since the boat is crossing the river at an angle, the component of the boat's velocity perpendicular to the river direction needs to be accounted for. This component is equal to the river current velocity. Therefore, we have:
V_perpendicular = C
Now we can find the speed of the boat in still water by considering the vector addition of the upstream and perpendicular components:
V = √((V_upstream)^2 + (V_perpendicular)^2)
Substituting the given values:
V = √((V * cos(45°))^2 + (2.3)^2)
To solve this equation, we can rearrange it and solve for V:
V^2 - (V * cos(45°))^2 = (2.3)^2
V^2 - V^2 * cos^2(45°) = 2.3^2
V^2 (1 - cos^2(45°)) = 2.3^2
V^2 * sin^2(45°) = 2.3^2
V^2 = (2.3^2) / sin^2(45°)
V = √((2.3^2) / sin^2(45°))
Calculating this value, we find:
V = 3.25 m/s
Therefore, the speed of the boat in still water is 3.25 m/s.
b) To find the time it takes for the boat to cross the river, we need to find the total distance the boat travels. This can be done by considering the horizontal component of the boat's velocity and the time it takes to cross the river.
Since the boat needs to cross a 260 m wide river and arrive at a point 110 m upstream, we can consider the total distance as the hypotenuse of a right triangle. Using the Pythagorean theorem:
Distance = √(260^2 + 110^2)
Calculating this value, we find:
Distance = √(67600 + 12100)
Distance = √79700
Distance ≈ 282.36 m
Now, considering the speed of the boat in still water, we can find the time it takes to cross the river:
Time = Distance / Speed
Time = 282.36 / 3.25
Calculating this value, we find:
Time ≈ 86.96 seconds
Therefore, the journey takes approximately 86.96 seconds.
To find the speed of the boat in still water and the time it takes for the journey, we can analyze the components of the boat's velocity.
Let's break down the velocity of the boat into its horizontal and vertical components.
Let V represent the speed of the boat in still water, and let φ represent the 45-degree angle the boat is heading upstream.
1) Speed of the boat in still water (V):
The horizontal component of the boat's velocity is the speed of the boat in still water, V. Since there is no current acting horizontally, the horizontal component of the boat's velocity remains constant throughout the journey.
So, the horizontal component of the boat's velocity is Vcos(φ).
2) Time taken for the journey (t):
To find the time it takes for the journey, we need to look at the vertical component of the boat's velocity, as the boat is moving upstream.
The vertical component of the boat's velocity (Vsin(φ)) determines how quickly the boat is crossing the river.
Now, let's calculate the values:
a) Speed of the boat in still water (V):
Since the boat must cross the river directly opposite its destination point, the vertical component of the boat's velocity is equal to the speed of the current.
Vsin(φ) = 2.3 m/s
b) Time taken for the journey (t):
The horizontal distance the boat must travel is the width of the river plus the distance upstream to the destination point, which is 260 m + 110 m = 370 m.
The vertical distance the boat must travel is the same as the width of the river, which is 260 m.
We can use the equation:
Distance = Speed × Time
For the horizontal distance:
370 m = Vcos(φ) × t (1)
For the vertical distance:
260 m = Vsin(φ) × t (2)
We know Vsin(φ) = 2.3 m/s from part (a).
Substitute Vsin(φ) = 2.3 m/s in equation (2) to solve for t:
260 m = 2.3 m/s × t
t = 260 m / 2.3 m/s
t ≈ 113.04 s
Now, substitute t = 113.04 s in equation (1) to solve for V:
370 m = Vcos(φ) × 113.04 s
V = 370 m / (cos(45°) × 113.04 s)
V ≈ 5.125 m/s
Therefore, the speed of the boat in still water (V) is approximately 5.125 m/s, and the time taken for the journey (t) is approximately 113.04 s.