Huck Finn walks at a speed of 0.60 m/s across his raft (that is, he walks perpendicular to the raft’s motion relative to the shore). The raft is traveling down the Mississippi River at a speed of 2.20 m/s relative to the river bank. What is Huck’s velocity (speed and direction) relative to the river bank?
To find Huck's velocity relative to the river bank, we need to consider the two velocities involved: Huck's velocity across the raft and the raft's velocity relative to the river bank.
Given:
Huck's velocity across the raft (v1) = 0.60 m/s
Raft's velocity relative to the river bank (v2) = 2.20 m/s
To find Huck's velocity relative to the river bank, we can use vector addition. We need to sum up the velocities vectorially.
Visualization: Imagine Huck standing on a moving raft (like a conveyor belt) while and he is walking across it.
If Huck walks across the raft at 0.60 m/s, we can represent this velocity as a vector pointing perpendicular to the direction of the raft's motion. Let's call this vector v1.
If the raft itself is moving downstream at 2.20 m/s, we can represent this velocity as a vector pointing downstream relative to the river bank. Let's call this vector v2.
Huck's velocity relative to the river bank (v3) is obtained by adding the two velocity vectors, v1 and v2, vectorially.
To add the vectors, we can use the Pythagorean theorem and trigonometry. The magnitude (speed) of Huck's velocity relative to the river bank is given by:
|v3| = √(v1^2 + v2^2)
Substituting the given values:
|v3| = √((0.60 m/s)^2 + (2.20 m/s)^2)
Simplifying:
|v3| = √((0.36 m^2/s^2 + 4.84 m^2/s^2)
|v3| = √(5.20 m^2/s^2)
|v3| ≈ 2.28 m/s
So, Huck's speed relative to the river bank is approximately 2.28 m/s.
To find the direction of Huck's velocity relative to the river bank, we can use trigonometry. The direction angle (θ) can be calculated using:
θ = arctan(v1/v2)
Substituting the given values:
θ = arctan(0.60 m/s / 2.20 m/s)
Calculating:
θ ≈ 15.4 degrees
This means that Huck's velocity relative to the river bank is at an angle of approximately 15.4 degrees with respect to the direction of the downstream flow of the river.