a motorboat is heading across a river at 6km/h in a direction making an angle 60 degree to the current. current is moving boat downstream at 3 km/h. what is the resultant velocity?
To find the resultant velocity of the motorboat, we can use the concept of vector addition. Let's break down the velocity of the motorboat into its horizontal and vertical components.
Given:
Speed of the motorboat (across the river) = 6 km/h
Angle between the boat's direction and the current = 60 degrees
Speed of the current (downstream) = 3 km/h
First, we need to find the horizontal and vertical components of the motorboat's velocity.
Horizontal Component:
The horizontal component is given by the product of the boat's speed and the cosine of the angle.
Horizontal Component = Speed of the boat * cos(angle)
Horizontal Component = 6 km/h * cos(60°)
Horizontal Component = 6 km/h * 0.5
Horizontal Component = 3 km/h
Vertical Component:
The vertical component is given by the product of the boat's speed and the sine of the angle.
Vertical Component = Speed of the boat * sin(angle)
Vertical Component = 6 km/h * sin(60°)
Vertical Component = 6 km/h * (√3/2)
Vertical Component = 3√3 km/h
Now, let's calculate the resultant velocity by adding the horizontal and vertical components.
Resultant Velocity (Magnitude):
Resultant Velocity = √(Horizontal Component^2 + Vertical Component^2)
Resultant Velocity = √(3^2 km/h + (3√3)^2 km/h)
Resultant Velocity = √(9 km²/h² + 27 km²/h²)
Resultant Velocity = √(36 km²/h²)
Resultant Velocity = 6 km/h
Resultant Velocity (Direction):
The resultant velocity has both a magnitude and a direction. The direction can be calculated using trigonometry. Since the boat is moving to the right (horizontal component) and downstream (vertical component), the angle can be found using the tangent function.
Angle = tan^(-1)(Vertical Component / Horizontal Component)
Angle = tan^(-1)((3√3 km/h) / 3 km/h)
Angle = tan^(-1)(√3)
Angle ≈ 60 degrees
Therefore, the resultant velocity of the motorboat is approximately 6 km/h at an angle of 60 degrees with the horizontal.