91Ó°ÊÓ

Displacement is the distance covered by a moving object in a specific direction along a straight line. In this exercise, we are given a displacement equation along an x-axis in the form of a polynomial expression: \[x = 18t + 5.0t^2\].This equation provides the position of the particle at any given time \(t\).

To perform a displacement calculation, you simply need to plug in the time \(t\) into the equation. For example, to find the displacement at \(t = 2.0\) seconds, substitute \(t = 2.0\) into the equation, which will yield:

• \( x(2.0) = 18 \times 2.0 + 5.0 \times (2.0)^2 = 36 + 20 = 56 \, \text{m} \) for \(t = 2.0\).
• Similarly, \( x(3.0) = 18 \times 3.0 + 5.0 \times (3.0)^2 = 54 + 45 = 99 \, \text{m} \) for \(t = 3.0\).

By substituting different values of \(t\), you can determine the displacement of the particle at various points in time.

Average velocity is a measure of the overall change in position over a specific time interval. It is calculated as the total displacement divided by the total time taken.

In this particular case, you are asked to find the average velocity between \(t = 2.0\) and \(t = 3.0\) seconds. The formula for average velocity \(\overline{v}\) is:\[ \overline{v} = \frac{x(t_2) - x(t_1)}{t_2 - t_1}\]For our values:

• Displacement at \(t_2 = 3.0\) seconds is \( x(3.0) = 99 \, \text{m} \).
• Displacement at \(t_1 = 2.0\) seconds is \( x(2.0) = 56 \, \text{m} \).

Plug these values into the formula:\[ \overline{v} = \frac{99 \, \text{m} - 56 \, \text{m}}{3.0 \, \text{s} - 2.0 \, \text{s}} = \frac{43 \, \text{m}}{1 \, \text{s}} = 43 \, \text{m/s}\]This calculation tells us that, on average, the particle is moving at a speed of 43 meters per second between these two time points.

Kinematics deals with the motion of objects without considering the forces that cause such motion. It involves concepts like displacement, velocity, and acceleration. In our exercise, we apply kinematic equations to understand how the particle moves along a straight line.

The kinematics of this problem begins by understanding the displacement equation: \( x = 18t + 5.0t^2 \). This provides a clear picture of the particle's position over time. To get more detailed insights into the particle's motion, kinematics tasks us with:

• Finding the velocity, by differentiating displacement with respect to time \(t\). The derivative, \( v(t) = \frac{dx}{dt} = 18 + 10t \), gives instantaneous velocity at any time \(t\).
• Instantaneous velocity reflects speed and direction at a specific moment, distinguishing it from average velocity, which covers an interval.
• Kinematics can extend further to assess other measures such as acceleration, which is the rate of change of velocity.

Understanding these fundamentals of kinematics provides a comprehensive foundation for solving various problems related to motion and speeds along a path.