91Ó°ÊÓ

Initial velocity is the speed and direction an object is moving at a starting point. In our exercise, it's given as a vector: \(\boldsymbol{v}_{1}=(6.30 \, \mathrm{m} / \mathrm{s}) \, \hat{\boldsymbol{i}} + (-8.42 \, \mathrm{m} / \mathrm{s}) \, \hat{\boldsymbol{j}}\). This vector tells us that the boat is moving 6.30 meters per second to the right (in the x-direction) and 8.42 meters per second downward (in the y-direction). It’s essential to understand both magnitude and direction to fully grasp initial velocity.

Final velocity is how fast an object is moving, and in which direction, at the end of a time interval. For the iceboat, the final velocity is zero in both directions: \(\boldsymbol{v}_{2} = 0 \, \mathrm{m} / \mathrm{s} \, \hat{\boldsymbol{i}} + 0 \, \mathrm{m} / \mathrm{s} \, \hat{\boldsymbol{j}}\). This means that the boat has come to a complete stop after 3 seconds due to the wind shift. Knowing the final velocity helps us find other quantities like acceleration and change in velocity.

Change in velocity is simply how much the velocity has changed over a period of time. It's calculated by subtracting the initial velocity from the final velocity: \(\boldsymbol{\triangle v} = \boldsymbol{v}_{2} - \boldsymbol{v}_{1}\). For our iceboat: \(\boldsymbol{\triangle v} = (0 \, \mathrm{m} / \mathrm{s} \, \hat{\boldsymbol{i}} + 0 \, \mathrm{m} / \mathrm{s} \, \hat{\boldsymbol{j}}) - (6.30 \, \mathrm{m} / \mathrm{s} \, \hat{\boldsymbol{i}} + (-8.42 \, \mathrm{m} / \mathrm{s}) \, \hat{\boldsymbol{j}}) = -6.30 \, \mathrm{m} / \mathrm{s} \, \hat{\boldsymbol{i}} + 8.42 \, \mathrm{m} / \mathrm{s} \, \hat{\boldsymbol{j}}\). This vector indicates how much and in what direction the boat's velocity has changed.

The time interval is the duration over which a change occurs. In our exercise, the time interval, \(\triangle t\), is 3 seconds. This interval is crucial for calculating average acceleration, which tells us how much the velocity changes per unit of time.

Vector components break down a vector into parts. For velocity, these are usually in the x and y directions. For our initial velocity, the vector components are 6.30 m/s in the x-direction and -8.42 m/s in the y-direction. For average acceleration: \(\boldsymbol{a}_{avg} = \frac{ -6.30 \, \mathrm{m} / \mathrm{s} \, \hat{\boldsymbol{i}} + 8.42 \, \mathrm{m} / \mathrm{s} \, \hat{\boldsymbol{j}} } {3 \, \mathrm{s}} = -2.10 \, \mathrm{m} / \mathrm{s}^2 \, \hat{\boldsymbol{i}} + 2.81 \, \mathrm{m} / \mathrm{s}^2 \, \hat{\boldsymbol{j}}\). Each component shows how different parts of the motion change over time.