Question

Given the position function \(\mathbf{r}\) of a moving object, explain how to find the velocity, speed, and acceleration of the object.

Short answer

Answer: Velocity is the rate at which the object's position changes with respect to time and is a vector, whereas speed is the magnitude of the velocity and is a scalar. To find the velocity, differentiate the position function \(\mathbf{r}(t)\) with respect to time: \(\mathbf{v}(t) = \frac{d\mathbf{r}(t)}{dt}\). To find the speed, calculate the magnitude of the velocity vector: \(s(t) = \sqrt{v_x^2(t) + v_y^2(t) + v_z^2(t)}\). To find the acceleration, differentiate the velocity function with respect to time: \(\mathbf{a}(t) = \frac{d\mathbf{v}(t)}{dt}\).

Step-by-step solution

1. Understanding Position Function, Velocity, Speed, and Acceleration

Position function, denoted as \(\mathbf{r}(t)\), represents the position of a moving object at time \(t\) in a given coordinate system. Velocity is the rate at which the object's position changes with respect to time, and speed is its magnitude. Acceleration is the rate at which the object's velocity changes with respect to time.

2. Finding the Velocity

In order to find the velocity of a moving object, we need to differentiate the position function \(\mathbf{r}(t)\) with respect to time. The velocity \(\mathbf{v}(t)\) is given by: $$\mathbf{v}(t) = \frac{d\mathbf{r}(t)}{dt}$$ This represents the first derivative of the position function with respect to time.

3. Finding the Speed

Speed is the magnitude of the velocity vector. To find the speed \(s(t)\), we need to calculate the magnitude of the velocity vector \(\mathbf{v}(t)\) that we found in the previous step. If the velocity vector has components \(\mathbf{v}(t) = \langle v_x(t), v_y(t), v_z(t)\rangle\), then the speed is given by: $$s(t) = \sqrt{v_x^2(t) + v_y^2(t) + v_z^2(t)}$$

4. Finding the Acceleration

To find the acceleration of the moving object, we need to differentiate the velocity function \(\mathbf{v}(t)\) with respect to time. Acceleration \(\mathbf{a}(t)\) is given by: $$\mathbf{a}(t) = \frac{d\mathbf{v}(t)}{dt}$$ This represents the second derivative of the position function \(\mathbf{r}(t)\) with respect to time, or the first derivative of the velocity function \(\mathbf{v}(t)\) with respect to time.

Key concepts

Position Function

Imagine following a moving object with your eyes as it travels through space. A position function encapsulates this visual path into a mathematical description. In calculus, the position function, often denoted as \(\mathbf{r}(t)\), describes the location of an object at any given time \(t\). Think of \(\mathbf{r}(t)\) as a snapshot of the object's coordinates in a specific coordinate system at each moment.

To improve understanding, envision the function as a trail on a map which exactly pinpoints where the object will be at various times throughout its journey. This function is essential in the study of motion as it serves as the foundation from which we derive more dynamic aspects of movement such as velocity and acceleration.

Velocity and Acceleration

With the path of an object defined by its position function, we now aim to comprehend how fast and in what direction it's moving. This is where velocity enters the scene. By taking the derivative of the position function \(\mathbf{r}(t)\) with respect to time, we obtain the velocity function, \(\mathbf{v}(t)\), which tells us the object's direction and rate of change in position at any moment in time. For clarity, if the position function represents the roadway, velocity is how fast you're driving on it and which way you're headed.

Speed, the Scalar Sister of Velocity

While velocity is a vector quantity including both magnitude and direction, speed is purely a measure of how fast the object is moving regardless of direction. To calculate speed, we assess the magnitude of the velocity vector, giving us a positive scalar value.

Acceleration further explores the rate at which velocity itself changes. It's the derivative of the velocity function with respect to time or, thinking two steps ahead, the second derivative of the position function. Therefore, acceleration tells us about the object's increasing or decreasing speed, as well as changes in direction.

Vector Calculus

Delving deeper into understanding the movement of an object, vector calculus is a pivotal tool. It’s a branch of mathematics concerned with differentiation and integration of vector fields, primarily in three-dimensional Euclidean space. Through vector calculus, we not only analyze the magnitude of vectors but also their directional components.

When we differentiate the position vector \(\mathbf{r}(t)\) to find velocity, or differentiate velocity to find acceleration, we are applying the fundamentals of vector calculus. It's like dissecting the path of a moving object into various segments to understand each tiny movement and its direction at every single point in time.

Enter the realm of multivariable calculus, where we often handle functions of several variables rather than just one. In such cases, the notion of partial derivatives becomes crucial. For a velocity vector \(\mathbf{v}(t) = \langle v_x(t), v_y(t), v_z(t)\rangle\), each component can be treated independently to capture the nuanced changes along each axis of the coordinate system.

Differential Calculus

To tie all this motion analysis together, we rely on the principles of differential calculus. This area of calculus focuses on the concept of the derivative, a measure of how a function changes as its inputs change. Essentially, it's the mathematics behind rates of change and slopes of curves.

Every time we discuss the rate of change of the position function to find velocity, or the rate at which velocity changes to find acceleration, we're doing differential calculus. It allows us to analyze motion in a detailed and precise manner, facilitating predictions about an object's future location and velocities based on its current state of motion.

• Derivative of position yields velocity.
• Derivative of velocity yields acceleration.
• Derivative of acceleration yields jerk, and so on.

By understanding differential calculus, students can grasp the infinite layers of change in moving objects, making it an invaluable tool for engineers, physicists, and mathematicians alike.