Suppose that the river was moving with a velocity of 3 m/s, North and the motor boat is moving with a velocity of 4 m/s, East. If the width of the river is 80 meters wide, then how much time does it take the boat to travel shore to shore? What distance downstream does the boat reach the opposite shore?
How long does it take the boat to cross the river of 80 m. wide at 4 m/s?
During this time, how far would the boat be carried downstream at 3 m/s?
To solve this problem, we can use vector addition to find the resultant velocity of the boat relative to the shore.
First, let's define the variables:
River velocity (Vr) = 3 m/s, North
Boat velocity (Vb) = 4 m/s, East
Width of the river (d) = 80 meters
To find the resultant velocity of the boat relative to the shore, we need to add the river velocity and the boat velocity vectorially. Since the river is flowing North and the boat is moving East, their velocities are perpendicular to each other.
Using the Pythagorean theorem, we can find the resultant velocity (V) of the boat relative to the shore:
V = √(Vr^2 + Vb^2)
The resultant velocity V is the speed at which the boat crosses the river. To find the time it takes for the boat to travel shore to shore, we can divide the width of the river by the resultant velocity:
Time (t) = d / V
To find the distance downstream that the boat reaches the opposite shore, we need to multiply the time by the river velocity:
Downstream distance = Vr * t
Let's calculate the values:
V = √(3^2 + 4^2) = √(9 + 16) = √25 = 5 m/s
t = d / V = 80 / 5 = 16 seconds
Downstream distance = Vr * t = 3 * 16 = 48 meters
Therefore, it takes the boat 16 seconds to travel shore to shore, and it reaches a point 48 meters downstream from the starting point on the opposite shore.