a kayaker paddles across a river at 4.2 m/s at an angle of 73 degrees northeast of the shoreline behind her. The river has a current of 2.4 m/s east acting separately from the motion of the kayaker. What is the kayaker's resultant velocity?
paddling across: 4.2cos73
paddling up river:4.2sin73
current down river:2.4
resultant velocity: sum of those
speed=sqrt(4.2^2cos^273+(4.2sin73-2.4)^2)
direction: arctan Ncomponent/Ecomponent
= arctan(4.2cos73/(4.2sin73-2.4) )
To find the kayaker's resultant velocity, we can use vector addition.
The kayaker's velocity is given as 4.2 m/s at an angle of 73 degrees northeast of the shoreline. This means that the kayaker is moving at a speed of 4.2 m/s in the northeast direction.
Separately, the river has a current of 2.4 m/s acting eastward.
To find the resultant velocity, we need to add the kayaker's velocity vector and the river's velocity vector.
First, we need to represent the kayaker's velocity vector and the river's velocity vector as components in the x and y directions.
The kayaker's velocity vector, at an angle of 73 degrees, can be broken down into its x and y components as follows:
Kayaker's velocity in the x direction = velocity * cos(angle) = (4.2 m/s) * cos(73 degrees) = 1.506 m/s
Kayaker's velocity in the y direction = velocity * sin(angle) = (4.2 m/s) * sin(73 degrees) = 3.908 m/s
The river's velocity vector is purely in the x direction, so the x component is equal to the river's velocity:
River's velocity in the x direction = 2.4 m/s
Now, we can add the x and y components separately to find the resultant velocity:
Resultant velocity in the x direction = Kayaker's velocity in the x direction + River's velocity in the x direction = 1.506 m/s + 2.4 m/s = 3.906 m/s
Resultant velocity in the y direction = Kayaker's velocity in the y direction = 3.908 m/s
So, the kayaker's resultant velocity is approximately 3.906 m/s in the x direction and 3.908 m/s in the y direction.