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Chapter 4: Problem 2

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

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

Step by step solution

01

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.
02

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 \( v_f = 0\hat{i} + 0\hat{j} \mathrm{m/s} \).
03

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: \( a = \frac{v_f - v}{t} \).
04

Calculate the Acceleration

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

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