91Ó°ÊÓ

In kinematics, the displacement function describes the position of a particle as it moves over time. For this exercise, the displacement function is given as: \[s(t) = t^3 - 6t^2 + 12t\] To solve problems involving displacement, we need to understand the function and how it changes with time. The function shows how far the particle is from its starting point at any given time t. Breaking it down, we have three main parts: - \(t^3\): This term illustrates the position's cubic growth or reduction over time. - \(-6t^2\): This term involves quadratic impact, possibly slowing down or speeding up the particle differently over time. - \(+12t\): This term adds a linear aspect to the particle's position. By combining these three, we get a comprehensive view of how the particle's position changes. Knowing the displacement function helps us find both velocity and acceleration, as these are derived from the displacement.

Velocity is the rate of change of displacement with respect to time. It tells us how fast and in which direction a particle is moving. From our displacement function, we calculate velocity by taking the first derivative. For the given function: \[s(t) = t^3 - 6t^2 + 12t\] The velocity is: \[v(t) = \frac{d}{dt}(t^3 - 6t^2 + 12t) = 3t^2 - 12t + 12\] Here, we used differentiation rules: - The derivative of \(t^3\) is \(3t^2\) - The derivative of \(-6t^2\) is \(-12t\) - The derivative of \(12t\) is \(12\) This function, \(v(t) = 3t^2 - 12t + 12\), now represents the velocity of the particle at any time t. Each component of the function tells us how quickly the displacement is changing at different moments. For example, \(3t^2\) shows rapid changes in velocity for larger values of t.

Acceleration is the rate of change of velocity with respect to time. It indicates how quickly the velocity changes. We find acceleration by taking the second derivative of the displacement function, which is the first derivative of the velocity function. From the given velocity function: \[v(t) = 3t^2 - 12t + 12\] The acceleration is: \[a(t) = \frac{d}{dt}(3t^2 - 12t + 12) = 6t - 12\] Here are the steps we follow to find it: - The derivative of \(3t^2\) is \(6t\) - The derivative of \(-12t\) is \(-12\) - The derivative of the constant \(12\) is \(0\) Therefore, the acceleration function is \(a(t) = 6t - 12\). This tells us how the velocity of the particle changes over time. For instance, at any time t, you can calculate the specific acceleration value to understand how the particle's speed is increasing or decreasing at that moment.