A rowboat is moving at 3 km/hr straight across a river from north to south and the river is flowing east at 4 km/hr. What is the resultant velocity?
X = 4 km/h
Y = 3 km/h
V = sqrt(X^2+Y^2)
To find the resultant velocity, we need to combine the velocity of the rowboat with the velocity of the river. The rowboat is moving at 3 km/hr straight across the river, which we can consider as the boat's "horizontal" velocity. The river is flowing east at 4 km/hr, which we can consider as the river's "vertical" velocity.
To combine these velocities, we can use vector addition. The rowboat's horizontal velocity and the river's vertical velocity can be thought of as two sides of a right-angled triangle. We can use the Pythagorean theorem to find the magnitude of the resultant velocity and trigonometry to find its direction.
The Pythagorean theorem states that the square of the hypotenuse (the resultant velocity) of a right-angled triangle is equal to the sum of the squares of the other two sides (horizontal and vertical velocities). In this case, the square of the resultant velocity (Vr^2) is equal to the square of the rowboat's horizontal velocity (Vb^2) plus the square of the river's vertical velocity (Vr^2):
Vr^2 = Vb^2 + Vr^2
Substituting the given values:
Vr^2 = (3 km/hr)^2 + (4 km/hr)^2
Vr^2 = 9 km^2/hr^2 + 16 km^2/hr^2
Vr^2 = 25 km^2/hr^2
Taking the square root of both sides of the equation:
Vr = √(25 km^2/hr^2)
Vr = 5 km/hr
So, the magnitude of the resultant velocity is 5 km/hr.
To find the direction of the resultant velocity, we can use trigonometry. We can consider the angle made by the resultant velocity with the horizontal direction as θ.
Using the definition of tangent:
tan(θ) = (Vertical Velocity) / (Horizontal Velocity)
tan(θ) = (4 km/hr) / (3 km/hr)
tan(θ) = 4/3
Now we can find θ using the arctangent function (tan^-1) on both sides of the equation:
θ = tan^-1(4/3)
Using a calculator, we find that θ is approximately 53.13 degrees.
Therefore, the resultant velocity is 5 km/hr at an angle of approximately 53.13 degrees with the horizontal direction.