You row a boat perpendicular to the shore of a river that flows at 2.5 m/s. The velocity of your boat is 4.5 m/s relative to the water.
(a) What is the velocity of your boat relative to the shore?
(b) What is the component of your velocity parallel to the shore?
(c) What is the component of your velocity perpendicular to the shore?
(a) v^2 = 2.5^2 + 4.5^2
(b) 2.5
(c) 4.5
(a) Well, if we combine the velocity of the river with the velocity of your boat, we get what we call the "velocity relative to the shore." So, get ready for the math magic! If we subtract the river's velocity (2.5 m/s) from your boat's velocity (4.5 m/s), we get a velocity relative to the shore of... drumroll please... 2 m/s!
(b) Ah, the parallel component! To figure that out, we need to break down your boat's velocity into two parts - the parallel and perpendicular components. Since we're interested in the parallel component, we can sneakily use some trigonometry. Considering that the direction of the river is perpendicular to the shore, the parallel component of your boat's velocity is 4.5 m/s * cos(90°) which is... 0 m/s! Can't go parallel if you're heading straight!
(c) Now it's time for the perpendicular component! Using a little trigonometric magic once again, we know that the perpendicular component of your boat's velocity is 4.5 m/s * sin(90°), giving us... ta-da! 4.5 m/s! So, hold on tight to that perpendicular ride!
To find the velocity of the boat relative to the shore, we can use vector addition.
(a) The velocity of the boat relative to the shore can be found by adding the velocity of the boat relative to the water and the velocity of the water.
Using vector addition, we can express this as:
Velocity of boat relative to shore = Velocity of boat relative to water + Velocity of water
Given:
Velocity of boat relative to water (Vb) = 4.5 m/s
Velocity of water (Vw) = 2.5 m/s
Using the vector addition, we can calculate it as:
Velocity of boat relative to shore = Vb + Vw
Velocity of boat relative to shore = 4.5 m/s + 2.5 m/s
Velocity of the boat relative to the shore = 7 m/s
Therefore, the velocity of the boat relative to the shore is 7 m/s.
(b) To find the component of velocity parallel to the shore, we can use the concept of trigonometry.
The component of velocity parallel to the shore is given by:
Velocity parallel to the shore = Velocity of the boat relative to shore × cos(θ)
where θ is the angle between the direction of the boat's velocity relative to the shore and the direction perpendicular to the shore.
Given:
Velocity of the boat relative to the shore = 7 m/s
To find θ, we have to note that the boat is rowed perpendicular to the shore. Therefore, the angle between the direction of the boat's velocity relative to the shore and the direction perpendicular to the shore is 90 degrees.
So, cos(90 degrees) = 0
Velocity parallel to the shore = 7 m/s × 0 = 0 m/s
Therefore, the component of the boat's velocity parallel to the shore is 0 m/s.
(c) To find the component of velocity perpendicular to the shore, we can once again use the concept of trigonometry.
The component of velocity perpendicular to the shore is given by:
Velocity perpendicular to the shore = Velocity of the boat relative to shore × sin(θ)
Since we already know the velocity of the boat relative to the shore is 7 m/s, we just need to find sin(θ).
Using the same angle θ as before (90 degrees), we have:
sin(90 degrees) = 1
Velocity perpendicular to the shore = 7 m/s × 1 = 7 m/s
Therefore, the component of the boat's velocity perpendicular to the shore is 7 m/s.
To find the answers to these questions, we can use vector addition and vector components. Let's break down the steps to get the answers:
(a) To find the velocity of your boat relative to the shore, we need to add the velocity of your boat relative to the water to the velocity of the water itself. Since the boat is rowing perpendicular to the shore, we can consider the velocity of the water as the velocity along the shore.
To find the velocity of your boat relative to the shore, we add the two velocities:
Velocity of the boat relative to the shore = Velocity of the boat relative to the water + Velocity of the water
Given:
Velocity of the boat relative to the water (vb) = 4.5 m/s
Velocity of the water (vw) = 2.5 m/s (since the river flows at a speed of 2.5 m/s)
So, Velocity of the boat relative to the shore = vb + vw = 4.5 m/s + 2.5 m/s = 7 m/s
Therefore, the velocity of your boat relative to the shore is 7 m/s.
(b) To find the component of your velocity parallel to the shore, we can consider the motion of the boat along the shore, which is perpendicular to the river's flow. In this case, the component of the boat's velocity parallel to the shore is the same as the velocity of the boat relative to the shore.
So, the component of your velocity parallel to the shore is 7 m/s.
(c) To find the component of your velocity perpendicular to the shore, we can consider the motion of the boat perpendicular to the shore, which is across the river's flow. In this case, the component of the boat's velocity perpendicular to the shore is the same as the velocity of the water.
So, the component of your velocity perpendicular to the shore is 2.5 m/s.