a)A man swims northeast at 5mph. Give a vector representing his velocity
b)He now swims a river the same velocity relative to still water; the river's current flows north at 1.2 mph. What is the velocity relative to the riverbed?
a) 5@045 OOO is N, O90 is E. or 5*.707N+5*.707E
b) 1.2N + 5*.707N+5*.707E= you do it.
a) Velocity is a vector quantity that represents both the speed and direction of an object's motion. In this case, the man is swimming northeast at a speed of 5 mph. To represent this velocity as a vector, we need to use the horizontal and vertical components of velocity.
To do this, we can break down the velocity into its horizontal and vertical components. The northeast direction can be split into the sum of the east and north directions, which are perpendicular to each other. Since the man is swimming at a constant speed of 5 mph, the magnitude of the velocity vector is 5 mph.
To find the horizontal and vertical components, we use trigonometry. Let's consider the angle between the northeast direction and the east direction as θ. Since the northeast direction is 45 degrees between the north and east, θ is 45 degrees.
The horizontal component can be found using the cosine function:
Horizontal Component = magnitude of velocity * cos(θ) = 5 mph * cos(45°)
The vertical component can be found using the sine function:
Vertical Component = magnitude of velocity * sin(θ) = 5 mph * sin(45°)
Using the values of cos(45°) = sqrt(2)/2 and sin(45°) = sqrt(2)/2, we can calculate:
Horizontal Component = 5 mph * sqrt(2)/2 = (5√2)/2 mph
Vertical Component = 5 mph * sqrt(2)/2 = (5√2)/2 mph
Therefore, the vector representing the man's velocity is (5√2)/2 mph east and (5√2)/2 mph north.
b) Now, considering the river's current flowing north at 1.2 mph, we need to find the velocity relative to the riverbed.
Since the man swims with the same velocity relative to still water, we can subtract the velocity of the current from the velocity of the man.
The horizontal component of the current is 0 mph (since it only flows north), and the vertical component is 1.2 mph.
To find the velocity relative to the riverbed, we subtract the current's velocity from the man's velocity:
Horizontal Component of Relative Velocity = Horizontal Component of Man's Velocity - Horizontal Component of Current's Velocity = (5√2)/2 mph - 0 mph
Vertical Component of Relative Velocity = Vertical Component of Man's Velocity - Vertical Component of Current's Velocity = (5√2)/2 mph - 1.2 mph
Therefore, the velocity relative to the riverbed is (5√2)/2 mph east and ((5√2)/2 - 1.2) mph north.