The position of an object as a function of time is \(\vec{r}=(3.2 t+\) \(\left.1.8 t^{2}\right) \hat{\imath}+\left(1.7 t-2.4 t^{2}\right) \hat{\jmath} \mathrm{m},\) with \(t\) in seconds. Find the object's acceleration vector.

Vector differentiation is a fundamental concept in physics and mathematics, particularly when dealing with motion. It involves finding the rate at which a vector-valued function changes with respect to one of its variables, commonly time.

Considering an object's position described by a vector that changes over time, to analyze its motion, we differentiate the position vector to obtain the velocity vector. This process is akin to finding the slope of the position-time graph at every instant, giving us the speed and direction of the object at that moment.

Understanding the Derivative of a Vector

When differentiating a vector, each component of the vector is differentiated separately. For instance, if a vector has components in the x and y directions, represented as \(\hat{i}\) and \(\hat{j}\), we differentiate the components associated with these unit vectors individually. This step-by-step process yields the velocity vector, which can be further differentiated to find the acceleration vector.

Applying this to the given problem, we started with the position vector and found the velocity vector. In the final step, we differentiated the velocity vector to provide the object's acceleration vector.

A position vector, denoted as \(\vec{r}\), represents an object's location in space relative to a fixed origin point. It is a fundamental concept in physics because it provides a way to quantify exactly where an object is at any given time.

For an object moving in two-dimensional space, its position vector can be expressed in the form \(\vec{r} = x(t)\hat{i} + y(t)\hat{j}\), where \(x(t)\) and \(y(t)\) are the coordinates of the object as functions of time, and \(\hat{i}\) and \(\hat{j}\) are the unit vectors in the horizontal and vertical directions, respectively.

Components of a Position Vector

Each term in the position vector contains a time-dependant part which gives us information on the trajectory of the object. In the exercise example, the object’s trajectory in space is a curve defined by the components \(3.2t + 1.8t^2\) in the direction of \(\hat{i}\) and \(1.7t - 2.4t^2\) in the direction of \(\hat{j}\). By analyzing the position vector, we can track the object as it moves, predicting its future positions and understanding its past.

The velocity vector \(\vec{v}\) is a key concept that arises from the differentiation of the position vector. It encapsulates not only how fast an object is moving but also the direction of its movement, making it a vector quantity.

The direction of the velocity vector is tangent to the path of motion at any given point, and its magnitude is the speed at which the object is traveling at that moment. Mathematically, we find the velocity vector \(\vec{v}\) by differentiating the position vector \(\vec{r}\) with respect to time.

Calculating the Velocity Vector

For our exercise, the velocity vector is found by differentiating the position vector's x and y components, yielding \(\vec{v} = (3.2 + 3.6t)\hat{i} + (1.7 - 4.8t)\hat{j}\). This result gives us an equation that describes how the velocity of the object changes over time. If the velocity vector components are constant, the movement is uniform. However, if the components change as a function of time, this indicates accelerated motion. In the given step-by-step solution, because the velocity vector's components are linear functions of time, they indicate that the object's motion is indeed accelerated.