91Ó°ÊÓ

Understanding how to calculate the velocity of a particle is crucial in mechanics. Velocity is defined as the rate of change of displacement with respect to time. In mathematical terms, we find the velocity by taking the first derivative of the displacement equation.

Let's look at our example, where the displacement is given by:


The displacement, \(s(t)\), of a particle is given by:


\[s(t) = t^3 - t^2 + 5\]

To find the velocity, denoted by \(v(t)\), we take the derivative of \(s(t)\) with respect to time \(t\). Differentiating each term with the power rule, we get:

\[v(t) = \frac{d}{dt}[t^3 - t^2 + 5]\]

Applying the power rule:

\[v(t) = 3t^2 - 2t\]

So, the velocity of the particle at any time \(t\) is \(3t^2 - 2t\). This equation tells us how fast the displacement is changing over time.

Acceleration is another fundamental concept that tells us how the velocity of a particle changes over time. Similar to velocity, which is the first derivative of displacement, acceleration is the first derivative of velocity, or equivalently, the second derivative of the displacement.

In our given example, after we have found the velocity equation:

\[v(t) = 3t^2 - 2t\]

To find the acceleration, represented as \(a(t)\), we differentiate the velocity equation with respect to time \(t\):

\[a(t) = \frac{d}{dt}[3t^2 - 2t]\]

Applying the power rule again, we get:

\[a(t) = 6t - 2\]

Thus, the acceleration of the particle at any time \(t\) is \(6t - 2\). This indicates how quickly the velocity of the particle is changing. If you substitute different values for \(t\) into this equation, you'll see how the acceleration varies with time.

Derivatives play a crucial role in mechanics and many other fields of science and engineering. They provide a way to calculate rates of change, which is essential for understanding physical phenomena like movement, growth, and decay.

In the context of our problem, we use derivatives to find both velocity and acceleration from the given displacement equation.
The steps are:

• Differentiate the displacement equation to find velocity.
• Differentiate the resulting velocity equation to find acceleration.

Given the displacement equation:

\[s(t) = t^3 - t^2 + 5\]

We take the first derivative to find velocity:

\[v(t) = 3t^2 - 2t\]

Then, we take the derivative of the velocity equation to find acceleration:

\[a(t) = 6t - 2\]

This process shows how powerful derivatives are in analyzing motion. They help us understand how quantities change and interact with each other over time.