A motor boat heads west across a river as a current of 12 km/hr. The speedometer reads a constant of 26 km/hr. What is the resultant velocity of the boat as compared to its original path? And what is the degree?
X = -26 km/h.
Y = -12 km/h(assuming a southbound current.
V^2 = X^2 + Y^2 = (-26)^2 + (-12)^2=820
Vb = 28.64 km/h.
tanAr = Y/X = -12/-26 = 0.46154
Ar = 24.8o. = Reference angle.
A = 180 + Ar = 180 + 24.8 = 204.8o,CCW.
= 24.8o South of West.
To find the resultant velocity of the boat as compared to its original path, we need to consider the vector addition of the boat's velocity and the current velocity.
Let's break down the problem into its components. We'll consider the boat's velocity as Vb and the current velocity as Vc.
The boat's velocity (Vb) can be determined by subtracting the current velocity (Vc) from the speedometer reading (26 km/hr). Since the boat is moving west, the boat's velocity is in the opposite direction of the current.
Speedometer reading (Vb) = 26 km/hr
Current velocity (Vc) = 12 km/hr
To find the resultant velocity, we subtract the current velocity (Vc) from the boat's velocity (Vb):
Resultant velocity (Vr) = Vb - Vc
Vr = 26 km/hr - 12 km/hr
Vr = 14 km/hr
The magnitude of the resultant velocity is 14 km/hr. This magnitude represents the speed at which the boat is moving with respect to its original path.
Now let's determine the degree. The degree refers to the angle between the boat's original path and its resultant path.
In this scenario, since the boat is moving west and the current is acting perpendicularly from north to south, the boat's original path is due west.
The resultant path can be represented as a diagonal vector resulting from the vector addition of the boat's velocity and the current velocity. Since the current acts in a perpendicular direction, the resultant path will be at an angle with respect to the boat's original path.
To find the angle between the original path and the resultant path, we can use trigonometry. We'll use the arctan function to find the angle.
Angle (θ) = arctan(Vc / Vb)
Angle (θ) = arctan(12 km/hr / 26 km/hr)
Using a calculator or trigonometric table, we find:
Angle (θ) ≈ 25 degrees
Therefore, the resultant velocity of the boat with respect to its original path is 14 km/hr, and the angle between the original path and the resultant path is approximately 25 degrees.