Problem 100 - Fundamentals of Physics

Question

An iceboat sails across the surface of a frozen lake with constant acceleration produced by the wind. At a certain instant the boat's velocity is \((6.30 \hat{\mathrm{i}}-8.42 \mathrm{j}) \mathrm{m} / \mathrm{s}\). Three seconds later, because of a wind shift, the boat is instantaneously at rest. What is its average acceleration for this \(3.00 \mathrm{~s}\) interval?

Step-by-Step Solution

Answer

The average acceleration is \(-2.10 \hat{\mathbf{i}} + 2.81 \hat{\mathbf{j}} \, \mathrm{m/s^2}\).

Step 1: Identify Initial and Final Velocities

The initial velocity of the ice boat is given as \( \mathbf{v_i} = 6.30 \hat{\mathbf{i}} - 8.42 \hat{\mathbf{j}} \, \mathrm{m/s} \). After 3 seconds, the final velocity is \( \mathbf{v_f} = 0 \hat{\mathbf{i}} + 0 \hat{\mathbf{j}} \, \mathrm{m/s} \) because the boat is at rest.

Step 2: Determine Time Interval

The time duration of the interval during which the boat comes to rest is \( t = 3.00 \mathrm{~s} \).

Step 3: Use the Average Acceleration Formula

Average acceleration \( \mathbf{a_{avg}} \) is calculated using the formula: \[ \mathbf{a_{avg}} = \frac{\mathbf{v_f} - \mathbf{v_i}}{t} \] where \( \mathbf{v_f} \) and \( \mathbf{v_i} \) are the final and initial velocities, respectively, and \( t \) is the time interval.

Step 4: Calculate Change in Velocity

Calculate the change in velocity: \[ \Delta \mathbf{v} = \mathbf{v_f} - \mathbf{v_i} = (0 \hat{\mathbf{i}} + 0 \hat{\mathbf{j}}) - (6.30 \hat{\mathbf{i}} - 8.42 \hat{\mathbf{j}}) = -6.30 \hat{\mathbf{i}} + 8.42 \hat{\mathbf{j}} \]

Step 5: Compute Average Acceleration

Substitute \( \Delta \mathbf{v} \) and \( t \) into the average acceleration formula: \[ \mathbf{a_{avg}} = \frac{-6.30 \hat{\mathbf{i}} + 8.42 \hat{\mathbf{j}}}{3.00} = -2.10 \hat{\mathbf{i}} + 2.81 \hat{\mathbf{j}} \, \mathrm{m/s^2} \]

Key Concepts

Vector Quantities

In physics, vector quantities are measurements that have both a magnitude and direction. Unlike scalar quantities, which have magnitude only, vectors require this additional directional component to fully describe them. This is crucial when studying motion, as many aspects such as velocity, acceleration, and force are vectors.

When dealing with vector quantities, you'll often encounter terms like the "i" and "j" components. These represent the x (horizontal) and y (vertical) directions, respectively. For example, in the problem provided, the iceboat's initial velocity is given in vector form as \(6.30 \hat{\mathbf{i}} - 8.42 \hat{\mathbf{j}}\ \textrm{m/s}\). This means the boat is moving positively in the x direction and negatively in the y direction at these respective rates.

• Magnitude: The size or length of the vector, calculated using the Pythagorean theorem: \( \sqrt{i^2 + j^2} \).
• Direction: The path that the vector points toward, which can be expressed using angles or simply by stating x and y components.
• Representation: Vectors are often illustrated using arrows on graph paper, where the length denotes magnitude and the arrow points in the direction of the vector.

Understanding vector quantities is key in solving physics problems involving motion, such as determining the iceboat's acceleration given its velocity changes.

Kinematics

Kinematics is the branch of physics that studies the motion of objects without considering the forces that cause this motion. It focuses on various parameters like velocity, acceleration, and displacement to describe how objects move.

In the given exercise, kinematics principles help us find the iceboat's average acceleration over a specific period. Key aspects to consider in kinematics are:

• Displacement: The change in position from the start to the end point, considered as a vector.
• Velocity: This vector quantity describes the rate of change of displacement over time. It includes both the speed and the direction of motion.
• Acceleration: The vector quantity representing the rate of change of velocity over time. In the problem, we calculated the average acceleration as \(-2.10 \hat{\mathbf{i}} + 2.81 \hat{\mathbf{j}}\ \textrm{m/s}^2\).

By using kinematic equations, especially when constant acceleration is involved, you can predict how an object will move. This enables solving the iceboat problem by systematically applying these fundamental equations.

Velocity Change

The concept of velocity change is pivotal in understanding motion. Velocity is not just about how fast an object is moving, but also the direction of its movement. Thus, any change in speed, direction, or both counts as a change in velocity.

In our iceboat problem, the change in velocity, \(\Delta \mathbf{v}\), is determined by subtracting the initial velocity from the final velocity. Given the final velocity is zero because the boat comes to a stop, the calculation simplifies to the negative of the initial velocity: \[\Delta \mathbf{v} = (0 \hat{\mathbf{i}} + 0 \hat{\mathbf{j}}) - (6.30 \hat{\mathbf{i}} - 8.42 \hat{\mathbf{j}}) = -6.30 \hat{\mathbf{i}} + 8.42 \hat{\mathbf{j}}.\]

• Magnitude of Change: Indicates how much the velocity has altered during the time interval.
• Direction of Change: Shows how the direction of the motion has shifted, which in this case, completely reverses due to a stop.

Understanding changes in velocity is critical because it directly impacts the acceleration—a core concept that describes how swiftly and in what manner motion changes.