A motor boat heads east across a river at 5.1 m/s. The river flows south at 2.1 m/s. Calculate the resultant speed.
v(R) =sqrt(5.1²+2.1²) = 5.52 m/s
To calculate the resultant speed, we need to find the vector sum of the eastward velocity of the boat and the southward velocity of the river.
To do this, we can use the Pythagorean theorem. The Pythagorean theorem states that the square of the hypotenuse (in this case, the resultant velocity) is equal to the sum of the squares of the other two sides.
In this case, the eastward velocity of the boat and the southward velocity of the river form a right triangle. The eastward velocity is the horizontal side, the southward velocity is the vertical side, and the resultant velocity is the hypotenuse.
Let's denote the eastward velocity of the boat as Veast and the southward velocity of the river as Vsouth.
Veast = 5.1 m/s (east)
Vsouth = 2.1 m/s (south)
Using the Pythagorean theorem:
Resultant velocity (Vresultant) = sqrt(Veast^2 + Vsouth^2)
Plugging in the given values:
Vresultant = sqrt((5.1 m/s)^2 + (2.1 m/s)^2)
= sqrt(26.01 m^2/s^2 + 4.41 m^2/s^2)
= sqrt(30.42 m^2/s^2)
≈ 5.51 m/s
Therefore, the resultant speed is approximately 5.51 m/s.