a powerboat heads due northwest at 15 m/s relative to the water across a river that flows due north at 4.8m/s what is the velocity of the motorboat and angle due west of north
To find the velocity of the powerboat and its angle due west of north, we can use vector addition.
First, let's break down the velocities into their horizontal and vertical components. The powerboat's velocity can be divided into two components: one along the northwest direction (let's call it Vnw) and the other perpendicular to it (let's call it Vp).
The northwest velocity (Vnw) is given as 15 m/s. Since velocity vectors add up, Vnw represents the combination of the boat's velocity relative to the water and the river's velocity. Since the river flows due north at 4.8 m/s, the boat's velocity relative to the water (Vwb) can be found by subtracting the river's velocity (Vr) from Vnw.
Vwb = Vnw - Vr
Vwb = 15 m/s - 4.8 m/s
Vwb = 10.2 m/s
Now, we need to determine the perpendicular component of the boat's velocity (Vp). This component is equal to the river's velocity since it flows northward. Thus, Vp = Vr = 4.8 m/s.
To find the resulting velocity and angle, we can use the Pythagorean theorem and trigonometry.
The resultant velocity (Vr) can be found by:
Vr = sqrt(Vwb^2 + Vp^2)
Vr = sqrt((10.2 m/s)^2 + (4.8 m/s)^2)
Vr ≈ 11.16 m/s
The angle can be found using the inverse tangent function:
θ = arctan(Vp / Vwb)
θ = arctan(4.8 m/s / 10.2 m/s)
θ ≈ 24.4 degrees
Therefore, the velocity of the powerboat across the river is approximately 11.16 m/s, and the angle it makes with due west of north is approximately 24.4 degrees.