Question

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

Short answer

The acceleration of the iceboat is \( a = -2.10\hat{i} + 2.81\hat{j} \mathrm{m/s^2} \).

Step-by-step solution

Identify Initial Velocity and Time

Firstly, we will denote the initial velocity vector as \( v = 6.30 \hat{i} - 8.42 \hat{j} \mathrm{m/s} \). The time it took for the boat to come to a stop is 3 seconds.

Set Up the Final Velocity Vector

The final velocity of the boat at the end of the 3-second interval is zero (the boat is said to be 'at rest'). So, we can denote the final velocity vector as \( v_f = 0\hat{i} + 0\hat{j} \mathrm{m/s} \).

Applying the Formula for Acceleration

Acceleration is defined as the change in velocity divided by the change in time. Using this definition, the acceleration vector \( a \) can be found using the formula: \( a = \frac{v_f - v}{t} \).

Calculate the Acceleration

Substitute the initial and final velocities, along with the time interval into the acceleration formula. Then simplify to find the acceleration: \( a = \frac{(0\hat{i} + 0\hat{j}) - (6.30\hat{i} - 8.42\hat{j})}{3s} = -2.10\hat{i} + 2.81\hat{j} \mathrm{m/s^2} \).

Key concepts

Constant Acceleration

Constant acceleration refers to a situation where an object's acceleration does not change over time. In simpler terms, the object's speed changes at a steady rate, whether increasing or decreasing. Imagine pressing the gas pedal in a car steadily so the increase in speed is smooth. This is analogous to constant acceleration. In physics problems, it is common to come across scenarios of constant acceleration, especially when dealing with motion in a straight line.

• This concept is crucial when analyzing motion because it allows us to predict how an object will move in the future.
• Constant acceleration simplifies calculations since the same formula can be applied throughout the motion.
• In the iceboat problem, the consistent acceleration due to wind allows us to use straightforward formulas to determine how the velocity changes over time.

Recognizing constant acceleration can simplify your calculations and help make predictions about subsequent movements with ease.

Velocity Vector

A velocity vector gives us not only the speed of an object but also the direction in which the object is moving. This is important because in physics, direction matters as much as speed. In the iceboat example, the initial velocity is given as a vector, which looks something like this: \(6.30 \hat{i} - 8.42 \hat{j}\, \mathrm{m/s}\).

• The 'i' and 'j' components represent motion along the x-axis and y-axis respectively.
• In layman's terms, the 'i' component can be thought of as movement east or west, and the 'j' component as movement north or south.
• Analyzing each component separately lets us understand the exact direction and path taken by an object.

Understanding velocity vectors is essential because they allow us to break down complex motion into simpler, one-dimensional movements that are easier to handle mathematically.

Acceleration Calculation

Acceleration calculation is a major part of understanding motion, as it tells us how quickly an object is speeding up or slowing down. In the iceboat scenario, we have to calculate the acceleration using the change in velocity over a specific time period. This gives us the acceleration vector, which indicates not only how much the velocity is increasing or decreasing but also in which direction these changes are occurring. To find the acceleration vector \( \mathbf{a} \), the formula is:\[\mathbf{a} = \frac{\mathbf{v}_f - \mathbf{v}}{t}\]Here:

• \( \mathbf{v}_f \) is the final velocity vector (\( 0\hat{i} + 0\hat{j} \, \mathrm{m/s} \)) after the boat stops.
• \( \mathbf{v} \) is the initial velocity vector (\( 6.30 \hat{i} - 8.42 \hat{j} \, \mathrm{m/s} \)).
• \( t \) is the time period (3 seconds).

After applying the values, the acceleration is found to be \(-2.10\hat{i} + 2.81\hat{j} \, \mathrm{m/s}^2 \). This implies the boat is decelerating in the direction of its initial velocity. Understanding this key process in physics allows us to apply similar calculations to other motion scenarios effectively.