a boat is traveling with a velocity of 42km\h(40 degree south of east) in a river. The river's current is 5km\h south. what will be the resultant velocity of the boat relative to the shore?
So I assume the 42km/hr is relative to the water?
If it is, relative to shore, you add the vectors
Draw the figure. The angle between the 42 and the 5 is 130 deg (check the figure). Law of Cosines:
speed^2=5^2+42^2 - 2(5)(43)cosine130
do the math
All anglesare measured CW from +y-axis
Vr = 42[130o] + 5[180o].
Vr = (42*sin130+5*sin180) + (42*cos130+5*cos180)I,
Vr = 32.2 - 32i = 45.4km/h[-45.2o] = 45.4km/h[45.2o] E. of S.
To determine the resultant velocity of the boat relative to the shore, we need to consider the vector addition of the boat's velocity and the river's current velocity.
First, let's break down the boat's velocity into its east and south components. The boat's velocity can be represented as 42 km/h at an angle of 40 degrees south of east. To find the east and south components, we can use trigonometric functions.
The east component of the boat's velocity can be calculated as follows:
East component = velocity * cos(angle)
East component = 42 km/h * cos(40 degrees)
Similarly, the south component of the boat's velocity can be calculated as follows:
South component = velocity * sin(angle)
South component = 42 km/h * sin(40 degrees)
Next, let's consider the river's current velocity. The current is flowing at 5 km/h south, so its south component will be -5 km/h, while the east component will be zero since the current flows straight down the river.
To find the resultant velocities, we can add the east and south components of the boat's velocity and the river's current velocity.
The east component of the resultant velocity is the sum of the boat's east component and the river's current east component. Since the river's current does not have an east component, the east component of the resultant velocity will be the same as the boat's east component.
East component of resultant velocity = East component of boat's velocity = 42 km/h * cos(40 degrees)
The south component of the resultant velocity is the sum of the boat's south component and the river's current south component.
South component of resultant velocity = South component of boat's velocity + South component of river's current
South component of resultant velocity = 42 km/h * sin(40 degrees) + (-5 km/h)
Finally, we can combine the east and south components of the resultant velocity to find the magnitude and direction of the resultant velocity (relative to the shore) using the Pythagorean theorem and trigonometry:
Resultant velocity = √[ (East component of resultant velocity)^2 + (South component of resultant velocity)^2 ]
Direction of resultant velocity (relative to the shore) = arctan[ (South component of resultant velocity) / (East component of resultant velocity) ]
By plugging in the values we calculated for the east and south components, you can find the magnitude and direction of the resultant velocity of the boat relative to the shore.