91Ó°ÊÓ

Velocity is a vector that describes the rate of change of position of an object over time. When you have a position function, like in our exercise \( \mathbf{r}(t) = (3t^2 + 1, 4t^2 + 3) \), the velocity function \( \mathbf{v}(t) \) is obtained by taking the derivative of the position function. This means finding how the position changes with respect to time.
In mathematics, differentiation provides this rate of change. Following the power rule, which states that the derivative of \( t^n \) is \( nt^{n-1} \), we can easily find the velocity. Differentiating the components of \( \mathbf{r}(t) \) gives us the velocity function: \( \mathbf{v}(t) = (6t, 8t) \). This tells us at any given time \( t \), how fast and in what direction the object is moving.
Importantly, because velocity is a vector, it contains both magnitude and direction. The first component \( 6t \) indicates movement along one axis, and the second component \( 8t \) along the other. Together, they give a full description of the object's motion in a plane.

Speed is a scalar quantity that tells us how fast an object is moving, without regard to direction. In mathematics, we calculate speed by finding the magnitude of the velocity vector, \( \mathbf{v}(t) \). This is done by taking each component of the velocity, squaring them, adding them together, and then taking the square root.
For the velocity vector \( \mathbf{v}(t) = (6t, 8t) \) from our exercise, the speed would be calculated as follows:

• Square each component: \( (6t)^2 \) and \( (8t)^2 \)
• Add them together: \( (6t)^2 + (8t)^2 = 36t^2 + 64t^2 = 100t^2 \)
• Take the square root: \( \sqrt{100t^2} = 10t \)

Thus, the speed of the object is \( 10t \). This value gives us a simple perspective on the rate at which the object is moving, without considering the direction of movement. Speed is always a non-negative value and is purely about how fast something goes.

Acceleration is a vector that indicates the rate of change of velocity over time. Just as velocity tells us how position changes, acceleration tells us how the velocity itself is changing. To find acceleration, we differentiate the velocity function with respect to time.
For our velocity function \( \mathbf{v}(t) = (6t, 8t) \), we apply the same power rule for differentiation:

• Differentiate \( 6t \) to obtain \( 6 \)
• Differentiate \( 8t \) to obtain \( 8 \)

As a result, the acceleration function is \( \mathbf{a}(t) = (6, 8) \). This vector says that for each unit of time, the velocity along one axis increases by 6 units, and along the other axis by 8 units.
Understanding acceleration helps us predict how an object's velocity changes, which is crucial in planning movements in engineering, physics, and real-world applications. Furthermore, like velocity, acceleration being a vector means both direction and magnitude play vital roles in its interpretation.