Question

The position vector \(r\) describes the path of an object moving in space. Find the velocity, speed, and acceleration of the object. $$ \mathbf{r}(t)=4 t \mathbf{i}+4 t \mathbf{j}+2 t \mathbf{k} $$

Short answer

The velocity of the object is \( \mathbf{v}(t) = 4 \mathbf{i} + 4 \mathbf{j} + 2 \mathbf{k} \), its speed is 6 units per time period, and it has no acceleration.

Step-by-step solution

Find the Velocity

Differentiate \( \mathbf{r}(t) = 4 t \mathbf{i} + 4 t \mathbf{j} + 2 t \mathbf{k} \) with respect to time to get the velocity vector \( \mathbf{v}(t) \). The result is \( \mathbf{v}(t) = 4 \mathbf{i} + 4 \mathbf{j} + 2 \mathbf{k} \).

Calculate the Speed

The speed is the magnitude of the velocity vector \( \mathbf{v}(t) = 4 \mathbf{i} + 4 \mathbf{j} + 2 \mathbf{k} \). The magnitude of this vector is \( | \mathbf{v}(t) | = \sqrt{(4)^2 + (4)^2 + (2)^2} = 6 \) units per time period.

Determine the Acceleration

The acceleration \( \mathbf{a}(t) \) is the derivative of the velocity vector with respect to time. It is found to be \( \mathbf{a}(t) = 0 \mathbf{i} + 0 \mathbf{j} + 0 \mathbf{k} \), since the coefficients of the velocity vector are constants.

Key concepts

Understanding Velocity Vector Calculus

The concept of a velocity vector in calculus is fundamental to understanding motion in space. Picture an object moving along a path; its position at any time can be represented by a position vector, denoted by r(t). But how fast is the object moving and in which direction?

The velocity vector, v(t), answers this. It is obtained by differentiating the position vector with respect to time. In our exercise, the differentiation of r(t) = 4ti + 4tj + 2tk gives us a constant velocity vector v(t) = 4i + 4j + 2k. This indicates that the object moves at a consistent speed and direction throughout its motion.

This concept is crucial, as velocity is not just about speed—it's also directional. Velocity vector calculus helps us visualize this motion in a three-dimensional space, which becomes essential in various fields, from physics to engineering.

Vector Differentiation

Vector differentiation is key when dealing with motion. It's a method used to find the rate at which the vector quantities change. In the context of motion, we differentiate position vectors to find velocity, and differentiate velocity vectors to find acceleration.

In the provided exercise, the process of vector differentiation was applied to the position vector to find the velocity. The derivatives of the vector's components, i, j, k, with respect to time t, are constants 4, 4, and 2, respectively.

Understanding vector differentiation requires familiarity with basic derivative rules from calculus. It’s important to apply these rules to each component of the vector separately. This approach allows us to derive the behavior of moving objects in a dynamic setting and creates a bridge for solving complex real-world problems involving motion.

Acceleration Vector Calculus

Acceleration is a measure of how quickly the velocity of an object changes with time. In vector calculus, the acceleration vector, a(t), is found by differentiating the velocity vector, v(t), with respect to time. It tells us about the object's change in speed and direction of travel.

In the exercise example, we determined that the velocity vector components are constants and so their derivative with respect to time is zero. Consequently, the acceleration vector a(t) = 0i + 0j + 0k suggests that there is no change in velocity over time; thus, the object moves with constant velocity. No acceleration implies no forces are acting on the object to change its state of motion, according to Newton's first law of motion.

This concept elucidates the relationship between an object's motion and forces acting on it, tying into Newtonian mechanics and allowing for the prediction of motion under different force conditions.