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The position-time relation is a fundamental concept in kinematics that describes how the position of a particle changes over time. This relation is important for analyzing motion along a line, such as the motion of a particle along the x-axis.
The position of a particle is often described using a function of time, such as in the equation given by the exercise: \(x(t) = 1 + t - t^2\). This equation gives the position \(x\) of the particle at any time \(t\). By substituting different values of \(t\) into the equation, we can determine where the particle is located at those times.
In the exercise, we calculated the initial position \(x(0)\) and final position \(x(2)\) to understand the particle's journey over a particular time interval.

Parabolic motion refers to any motion following a path that forms a parabola. In this exercise, the function describing the particle's movement \(x(t) = 1 + t - t^2\) is a quadratic equation, indicating that the motion path is a parabola.
Parabolas open either upwards or downwards depending on the sign of the quadratic term (\(t^2\)). Here, the parabola opens downwards because the coefficient of \(t^2\) is negative (\(-1\)). Understanding the nature of the parabola gives insight into the change of direction of the particle. The particle will reach its maximum position at the vertex, and then reverse direction.
This allows us to predict the general motion and anticipate key points in the particle's journey, such as turning points or maximum height.

Calculating the distance travelled by a particle involves evaluating the total length of the path it covers, regardless of changes in direction. It is crucial to consider every segment of the journey. This is particularly important when dealing with parabolic motion, where a simple calculation by subtracting final and initial positions would not suffice.
For example, in finding the distance traveled between \(t = 0\) and \(t = 2\), we need to account for the vertex, where the direction changes. We calculate the partial distances from \(x = 1\) to the vertex \(x = 1.25\) and from the vertex to the final position \(x = -1\).
Adding these segments together yields the total distance: \(0.25\) m from start to vertex and \(2.25\) m from vertex to end, totaling \(2.5\) m.

In the context of motion, the vertex of a parabola is a critical point. It represents the peak or trough of the trajectory, where the object changes direction. For the equation \(x(t) = 1 + t - t^2\), finding the vertex helps us understand exactly when and where this change occurs.
To find the vertex in a parabolic equation \(at^2 + bt + c\), use the formula \(t = -\frac{b}{2a}\). Applying this to our equation with \(a = -1\) and \(b = 1\), we calculate \(t = \frac{1}{2}\). At this point, we plug it back into the position function to find \(x(0.5) = 1.25\).
Knowing the vertex allows us to accurately determine the distance the particle travels before and after reaching this peak, ensuring precise distance calculations.