Question

A ferry approaches shore, moving north with a speed of \(6.2 \mathrm{~m} / \mathrm{s}\) relative to the dock. A person on the ferry walks from one side of the ferry to the other, moving east with a speed of \(1.1 \mathrm{~m} / \mathrm{s}\) relative to the ferry. What is the speed of the person relative to the dock?

Short answer

The speed of the person relative to the dock is approximately 6.3 m/s.

Step-by-step solution

Understanding the Problem

We need to find the resultant velocity of a person moving in two perpendicular directions. The ferry moves north at a speed of \(6.2 \; \text{m/s}\), and the person walks east at a speed of \(1.1 \; \text{m/s}\) on the ferry. These can be considered as two components of the person's velocity relative to the dock.

Identify the Components of Velocity

We'll consider the northward velocity \(v_{y} = 6.2 \; \text{m/s}\) and the eastward velocity \(v_{x} = 1.1 \; \text{m/s}\). These velocities are perpendicular, with the ferry's motion representing the north-south component and the person's walking representing the east-west component.

Apply the Pythagorean Theorem

To find the resultant speed relative to the dock, we use the Pythagorean theorem because the velocity components form a right triangle. The formula is: \[ v = \sqrt{v_{x}^2 + v_{y}^2} \]Substitute the values: \[ v = \sqrt{(1.1)^2 + (6.2)^2} \]

Calculate the Resultant Velocity

Perform the calculation: \[ v = \sqrt{1.21 + 38.44} = \sqrt{39.65} \approx 6.295 \; \text{m/s} \] The speed of the person relative to the dock is approximately \(6.3 \; \text{m/s}\).

Key concepts

Resultant Velocity

In the context of physics, especially when discussing relative motion, the concept of resultant velocity is critical. It refers to the combination of two or more velocities into a single velocity, which effectively represents the direction and speed with which an object moves relative to a particular frame of reference.

To find the resultant velocity, we often look at situations where different velocity vectors are acting simultaneously. In the case of the ferry problem, two distinct velocities affect the person walking on it: the velocity of the ferry moving north and the velocity of the person moving east. By combining these two vectors, we aim to find out how the person's movement appears from a stationary observer standing on the dock. This resultant velocity provides a complete picture of the person's movement as seen from the dock.

• Resultant velocity considers direction and magnitude.
• It is computed by vector addition of individual velocities.
• Key in determining the actual path and speed of moving objects relative to each other.

Pythagorean Theorem

The Pythagorean Theorem is a mathematical principle that links the lengths of the sides of a right triangle. This theorem is vital in physics for solving problems involving perpendicular vectors. Given a right triangle, the 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. The formula is expressed as \[ c^2 = a^2 + b^2 \].

In the case of the ferry exercise, the theorem helps us derive the resultant velocity by treating the northward and eastward movements as two sides of a right triangle. By using the Pythagorean theorem, we can find the hypotenuse, which represents the resultant velocity of the person as observed from the dock.

• Useful for finding the magnitude of resultant vectors.
• Applies to any scenario involving right-angled vector contributions.
• Helps to solve real-life physics problems where vectors form perpendicular angles.

Velocity Components

When analyzing motion, especially in two dimensions, breaking down a vector into its components helps simplify the problem. Each component represents the vector's influence in one particular direction, typically aligned with the coordinate axes, like north-south and east-west.

In the ferry scenario, the velocity vector of the person can be split into two components:

• The northward velocity component \(v_y = 6.2 \; \text{m/s}\)
• The eastward velocity component \(v_x = 1.1 \; \text{m/s}\)

These components are perpendicular to each other and can be individually measured in terms of their magnitude and direction.

Understanding velocity components is crucial because it allows for easy calculation of the resultant velocity through vector addition. By solving for each component separately, we gain insights into how each direction contributes to the overall motion. This technique is invaluable in physics to simplify complex three-dimensional motion into more manageable two-dimensional problems.