91Ó°ÊÓ

Velocity measures how fast something moves in a specific direction. In this exercise, we are given a velocity vector that depends on time. The vector is \( \vec{v} = (6.0t - 4.0t^2) \hat{\mathrm{i}} + 8.0 \hat{\mathrm{j}} \), where \( \hat{\mathrm{i}} \) and \( \hat{\mathrm{j}} \) represent direction in the plane. This velocity tells us both the speed and direction of particle movement.
Understanding the components of velocity:

• **The \( \hat{\mathrm{i}} \) component** varies with time, indicating that the speed in the x-direction changes as time progresses.
• **The \( \hat{\mathrm{j}} \) component** is constant at 8.0 m/s. This component reveals a steady speed in the y-direction.

Analyzing this setup helps us see how velocity can change not only in magnitude but also in direction over time. This is why knowing both components is crucial to fully describing motion in two dimensions.
When examining conditions such as when the velocity is zero, both components must be zero simultaneously. However, since our j-component is constant, the velocity vector never truly reaches zero.

Acceleration describes the rate at which velocity changes over time. In our given exercise, to find acceleration, we need to differentiate the velocity vector concerning time.
Here's what you need to understand about acceleration:

• **Differentiating the i-component of velocity** \(6.0t - 4.0t^2\) leads to the i-component of acceleration: \(6.0 - 8.0t\).
• **Differentiating the constant j-component** (8.0) gives 0, meaning there's no change in speed in the y-direction.

The composite acceleration vector comes out to be \(\vec{a} = (6.0 - 8.0t) \hat{\mathrm{i}}\). Notice how the y-component is absent, indicating no acceleration in that direction.
Finding when acceleration is zero requires setting the i-component of acceleration to zero, which occurs when \( t = 0.75 \) seconds. This signifies a moment when the particle changes its speed but not its direction.

Differentiation is a mathematical process used to determine how a function changes as its input changes. Here, it's key in switching from knowing just the velocity to understanding the acceleration.
For the exercise:

• We differentiate each component of the velocity vector to derive the respective components of the acceleration vector.
• Observing the constant explains why its derivative is zero—it doesn't change with time, hence no resulting acceleration in its direction.

Through differentiation, you also discern critical conditions like when quantities such as velocity or acceleration reach specific values. For example, knowing how velocity's components depend on time lets you solve when its magnitude matches a predetermined speed, such as 10 m/s in this problem. Differentiation allows us to translate the behavior of velocity into the behavior of acceleration efficiently.