A boat crossing a .2 mile wide river is heading perpendicularly relative to shore. Its speed in still water is 4mph and the speed of the current is 6mp.
a.) How far downstream will the boat land?
b.) How long will it take for the boat to cross
i.) with the current?
ii.) without the current?
i) a. 3 mile b. 0.5 hr ii)a. 0 mile ( no deflection distance) b) 0.2774 hr
To find the answers to these questions, we can apply the concept of relative velocity and use some basic trigonometry. Let's break it down step by step:
a) How far downstream will the boat land?
To determine how far downstream the boat will land, we need to find the horizontal component of the boat's velocity.
First, we need to calculate the effective velocity of the boat with respect to the ground. Since the boat is moving perpendicular to the shore, the effective velocity can be found using the Pythagorean theorem:
Effective velocity = √(velocity in still water)^2 + (velocity of current)^2
Plugging in the values given:
Effective velocity = √(4 mph)^2 + (6 mph)^2
Effective velocity ≈ √16 + 36 ≈ √52 ≈ 7.21 mph
Now, we can calculate how long it will take for the boat to cross the river:
Time = Distance / Effective velocity
The distance across the river is given as 0.2 miles, and the effective velocity of the boat is approximately 7.21 mph.
Plugging in the values:
Time = 0.2 miles / 7.21 mph ≈ 0.028 hours
To find how far downstream the boat will land, we use the formula:
Distance downstream = (velocity of current) × Time
Plugging in the values:
Distance downstream = 6 mph × 0.028 hours
Distance downstream ≈ 0.17 miles
Therefore, the boat will land approximately 0.17 miles downstream.
b) How long will it take for the boat to cross?
i) With the current:
We already calculated the time it takes for the boat to cross the river with the current in part a). It is approximately 0.028 hours.
ii) Without the current:
To find the time it takes for the boat to cross the river without the current, we need to calculate the effective velocity in the opposite direction, which is the velocity in still water minus the velocity of the current:
Effective velocity (opposite direction) = (velocity in still water) - (velocity of current)
Effective velocity (opposite direction) = 4 mph - 6 mph = -2 mph
Since the velocity in the opposite direction is negative, we take the absolute value to get the magnitude of the effective velocity.
Effective velocity (opposite direction) = 2 mph
Using the same formula as before:
Time = Distance / Effective velocity
We plug in the value of distance (0.2 miles) and the magnitude of the effective velocity (2 mph):
Time = 0.2 miles / 2 mph = 0.1 hours
Therefore, it will take approximately 0.1 hours (or 6 minutes) for the boat to cross the river without the current.