Question

The position vector \(r\) describes the path of an object moving in space. Find the velocity, speed, and acceleration of the object. $$ \mathbf{r}(t)=\langle 4 t, 3 \cos t, 3 \sin t\rangle $$

Short answer

The velocity of the object is \(\langle4, -3 \sin t, 3 \cos t \rangle\), the speed is \(5\), and the acceleration is \(\langle0, -3 \cos t, -3 \sin t\rangle\).

Step-by-step solution

Differentiate the Position Vector to find Velocity

To find the velocity of the object, differentiate each component of the position vector with respect to \(t\). Hence, velocity \(v(t)\) is \(\langle4, -3 \sin t, 3 \cos t \rangle\).

Find the Magnitude of the Velocity Vector to get Speed

The speed of the object is the magnitude of the velocity vector. To find this, square each component of the vector, sum them up and take the square root, giving \(\sqrt{{4}^2+(-3 \sin t)^2+(3 \cos t)^2}\) which simplifies to \(5\).

Differentiate the Velocity Vector to find Acceleration

Finally, to find the acceleration, differentiate the velocity vector with respect to \(t\). This gives acceleration \(a(t)\) as \(\langle0, -3 \cos t, -3 \sin t\rangle\).

Key concepts

Velocity Vector

Understanding the concept of the velocity vector is foundational in vector calculus, particularly when describing the motion of objects. This vector represents not only how fast an object is moving, but also in what direction it's heading at any given moment in time. To find the velocity vector, we differentiate the position vector with respect to time.

In our exercise, by differentiating the position vector \(\textbf{r}(t)=\textless 4t, 3 \text{cos} t, 3 \text{sin} t\textgreater\), we obtained the velocity vector \(v(t)=\textless 4, -3 \text{sin} t, 3 \text{cos} t\textgreater\). Each component of this vector provides information about the speed of the object along the corresponding axis.

The velocity vector is crucial for understanding motion because it encapsulates the rate of change of the object’s position. It's often one of the first steps toward deeper analysis in kinematics and dynamics, and it lays the groundwork for concepts like acceleration and force.

Speed of an Object in Motion

The speed of an object in motion is a scalar quantity that simply measures how fast an object is moving irrespective of its direction. In contrast to the velocity vector, speed does not provide directional information, thus it's always a positive value.

To determine the speed from the velocity vector, we calculate the vector's magnitude. This step can be thought of as the Pythagorean theorem in higher dimensions. For our exercise, we take the magnitude of the velocity vector \(v(t)=\textless 4, -3 \text{sin} t, 3 \text{cos} t\textgreater\) resulting in \(\text{speed} = \text{magnitude of } v(t) = \text{square root of sum of the squares of the components}\), which simplifies to a constant speed of 5 units.

Why is speed constant here?

This uniform speed indicates that the object's rate of travel through space is steady, not accelerating nor decelerating in the context of speed as a scalar. Understanding speed is vital as it helps to conceptualize the intensity of motion in practical situations, such as when calculating travel times or when analyzing objects moving at constant speeds.

Acceleration Vector

The acceleration vector is another fundamental concept in vector calculus that describes how an object’s velocity changes over time. Just like velocity, acceleration has both magnitude and direction. It is found by differentiating the velocity vector with respect to time.

In our given problem, we obtain the acceleration vector by differentiating \(v(t)=\textless 4, -3 \text{sin} t, 3 \text{cos} t\textgreater\), resulting in \(a(t)=\textless 0, -3 \text{cos} t, -3 \text{sin} t\textgreater\). This indicates how the object's velocity changes at every instant, which is essential in predicting future paths and forces required to alter motion.

Interpreting Zero Acceleration

Notably, the first component of the acceleration vector is zero, which tells us that there’s no change in velocity along the x-axis, and thus no acceleration in that direction. Understanding the acceleration vector can be crucial in various fields such as mechanical engineering, aeronautics, and any domain where the dynamics of moving objects are relevant.