Problem 58 - Physics for Scientists and Engineers with Modern Physics

Question

Huck Finn walks at a speed of \(0.70 \mathrm{~m} / \mathrm{s}\) 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 \(1.50 \mathrm{~m} / \mathrm{s}\) relative to the river bank (Fig. \(3-49\) ). What is Huck's velocity (speed and direction) relative to the river bank?

Step-by-Step Solution

Answer

Huck's velocity relative to the river bank is 1.66 m/s at 25.0° from the raft's direction.

Step 1: Understand the Concept

To solve this problem, we need to use vector addition. Huck's velocity relative to the river bank is the vector sum of his velocity relative to the raft and the raft's velocity relative to the river bank.

Step 2: Define the Velocities

Huck's walking speed relative to the raft is given as \(0.70 \text{ m/s}\). The raft's speed relative to the river bank is \(1.50 \text{ m/s}\). Huck walks perpendicular to the raft's motion, forming a right triangle.

Step 3: Apply the Pythagorean Theorem

Since Huck walks perpendicular to the raft's motion, we can use the Pythagorean theorem to find his resultant speed relative to the river bank: \( v = \sqrt{(0.70)^2 + (1.50)^2} \).

Step 4: Calculate the Magnitude

Calculate the magnitude using the equation: \( v = \sqrt{(0.70)^2 + (1.50)^2} = \sqrt{0.49 + 2.25} = \sqrt{2.74} \approx 1.66 \text{ m/s}\).

Step 5: Determine the Direction

The direction can be determined using trigonometry. The angle \( \theta \) with respect to the river bank's direction (raft's direction) is given by \( \theta = \tan^{-1}\left(\frac{0.70}{1.50}\right) \).

Step 6: Calculate the Direction Angle

Calculate the angle \( \theta \): \( \theta = \tan^{-1}\left(\frac{0.70}{1.50}\right) = \tan^{-1}(0.467) \approx 25.0^\circ \). Huck's velocity is 1.66 m/s at an angle of 25.0 degrees from the raft's direction.

Key Concepts

Relative Velocity

In the context of physics, relative velocity is a core concept when evaluating how the speed of one object is observed concerning another. In our scenario with Huck Finn, we're assessing how his movement relative to the raft combines with the raft's own motion to exhibit a total speed and direction relative to the river bank. This is crucial because:

• Each object has its own velocity (speed and direction) relative to another.
• By using vector addition, we can find the resultant velocity, which gives us Huck's effective speed and direction relative to the shore.
• Visualizing motion in this composite manner helps to solve problems where objects are in motion from different frames of reference.

Understanding relative velocity is akin to standing on a moving walkway at the airport: your personal walking speed combines with the walkway's speed to give your total travel speed relative to someone standing still.

Pythagorean Theorem

The Pythagorean Theorem is a mathematical principle that allows us to relate the sides of a right triangle. If you remember, this theorem states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.In Huck's case:

• Huck's walking direction and the raft's motion form a right triangle.
• The Pythagorean Theorem is used to find the resultant speed, as Huck's path and the raft's path are perpendicular.

By applying this theorem, we calculate the magnitude of Huck's velocity relative to the river bank. The equation is straightforward: \[ v = \sqrt{(0.7)^2 + (1.5)^2} \]This equation helps us combine Huck's walking speed and the raft's speed into a single measure, showing how his total motion relates to the shore.

Trigonometry

Trigonometry might seem complex, but it's pivotal in understanding movements involving angles and directions. It helps us determine the direction of objects that are moving along paths not aligned with standard horizontal or vertical axis.In finding Huck’s direction relative to the riverbank:

• We use the tangent function to relate the angle of Huck's trajectory with respect to the raft's motion.
• This involves the ratio of the perpendicular side (Huck's walking speed) to the base (raft's speed), as given by \( \theta = \tan^{-1}\left(\frac{0.7}{1.5}\right) \).
• This angle indicates how much Huck's path deviates from the raft's direction, resulting in a combined vector path.

By understanding these fundamentals of trigonometry, we turn a potentially confusing aspect of physics into a precise calculation, allowing us to fully grasp the direction in a movement scenario like Huck's.