a boat crosses a river and arrives at a point on the opposite bank directly across from its starting position. The boat traveks at 4m/s and the current is 2m/s. If the river is 600 m wide in what direction must the boat steer and how long will it take to cross?
current toward -y
We want to go in +x direction
steer at angle T up from x axis
components toward x
Vx = 4 cos T
components toward y
Vy = 4 sin T - 2
we want Vy = 0
so
2 = 4 sin T
sin T = 1/2
T = 30 degrees or in compass heading
30 degrees North of East = 60 degrees true
To determine the direction in which the boat must steer and how long it will take to cross the river, we need to analyze the situation.
Let's assume the river flows horizontally from left to right. The boat's velocity relative to the water is 4 m/s, and the current is flowing with a velocity of 2 m/s in the same direction. Therefore, the total velocity of the boat relative to the ground is the vector sum of its velocity relative to the water and the velocity of the current.
The boat wants to arrive directly across from its starting position, so it needs to steer at an angle upstream to counteract the effect of the current pulling it downstream. Let's call this angle θ.
To determine the magnitude of the boat's velocity relative to the ground, we can use the Pythagorean theorem. The width of the river is given as 600 m, and during the crossing, the boat has to travel this distance. Using the formula:
(v_relative_to_ground)^2 = (v_relative_to_water)^2 + (v_of_current)^2
We have:
(v_relative_to_ground)^2 = (4 m/s)^2 + (2 m/s)^2
(v_relative_to_ground)^2 = 16 m^2/s^2 + 4 m^2/s^2
(v_relative_to_ground)^2 = 20 m^2/s^2
v_relative_to_ground = √20 m/s
Next, we can use trigonometry to find the direction in which the boat must steer. The tangent of angle θ can be calculated using the formula:
tan(θ) = (opposite)/(adjacent)
tan(θ) = (v_of_current)/(v_relative_to_water)
tan(θ) = 2 m/s / 4 m/s
tan(θ) = 0.5
To find θ, we can take the inverse tangent (tan^(-1)) of 0.5:
θ = tan^(-1)(0.5)
θ ≈ 26.57°
Therefore, the boat must steer at an angle of approximately 26.57° upstream from straight across the river.
Now we can calculate the time it takes for the boat to cross the river. The boat's velocity relative to the ground is √20 m/s, and the width of the river is 600 m. Time (t) can be calculated using the formula:
time = distance / velocity
t = 600 m / √20 m/s
t ≈ 66.67 seconds
So, the boat will take approximately 66.67 seconds to cross the river.