The position vector r describes the path of an object moving...

The position vector $r$ describes the path of an object moving in the $x y$ -plane. (a) Find the velocity vector, speed, and acceleration vector of the object. (b) Evaluate the velocity vector and acceleration vector of the object at the given point. (c) Sketch a graph of the path, and sketch the velocity and acceleration vectors at the given point. $$\begin{array}{lll}\text { Position Vector } & \text { Point } \\ \mathbf{r}(t)=3 t \mathbf{i}+(t-1) \mathbf{j} & \quad (3,0)\end{array}$$

The position vector $r$ describes the path of an object moving in the $x y$ -plane.
(a) Find the velocity vector, speed, and acceleration vector of the object.
(b) Evaluate the velocity vector and acceleration vector of the object at the given point.
(c) Sketch a graph of the path, and sketch the velocity and acceleration vectors at the given point.
$$\begin{array}{lll}\text { Position Vector } & \text { Point } \\ \mathbf{r}(t)=3 t \mathbf{i}+(t-1) \mathbf{j} & \quad (3,0)\end{array}$$

Key Concepts

Evaluation at Specific Points

Evaluating vector functions like velocity and acceleration at specific points involves substituting a particular value of the parameter (often time) into these functions. This allows for the analysis of the object's instantaneous state of motion at that given point.

Graphical Representation of Motion

Graphing the position vector aids in visualizing the trajectory of the object in space. Additionally, sketching the velocity and acceleration vectors at a point helps in understanding the dynamics of the object's motion by illustrating the direction and magnitude of these quantities at that particular instance.

Acceleration Vector

The acceleration vector is defined as the derivative of the velocity vector with respect to time. It describes how the velocity changes over time, capturing both changes in speed and changes in direction, and is central to understanding an object's kinematics.

Position Vector and Parametric Equations

The position vector provides a way to represent the location of an object in space as a function of a parameter, usually time. It is expressed in component form (typically using i, j, and possibly k for three dimensions) and describes the path of the object in the coordinate plane, forming the basis for further analysis of motion.

Velocity Vector and Derivatives

The velocity vector is the first derivative of the position vector with respect to time. It indicates both the speed and the direction of the object's motion. Differentiation of vector functions is a fundamental concept that allows us to determine how an object's position changes over time.

Speed as the Magnitude of the Velocity Vector

Speed is defined as the magnitude of the velocity vector, providing a scalar measure of how fast the object is moving regardless of its direction. It is computed by taking the square root of the sum of the squares of the velocity components.

Transcript

00:01 Okay, we're given a position vector, and the first thing that we are going to do is we're going to find its velocity vector.

00:10 So in order to do that, we'll take the derivative of our position vector.

00:16 So to take the derivative, we take the derivative of each individual piece.

00:20 So the derivative of 3t is 3.

00:24 So there's 3 in the i direction, and then plus the derivative of t minus 1 is just 1j.

00:33 So we know our speed without actually even finding a time to put in.

00:39 Normally, you would still have t's in your velocity, but both were linear.

00:44 So we took our 3 squared plus our 1 squared and we get 10 and then we take the square root of 10.

00:51 So the speed is the square root of 10.

00:54 Now, acceleration will be the derivative of velocity, but each of the components are a actual you know, it's just a value.

01:03 There's no more variables left, and so our acceleration is zero in both directions.

01:10 Now, normally, you would need to have already done this piece, but what we need to do is we need to set, you know, the point is given in terms of x and y.

01:21 Well, our x piece is 3t.

01:24 So if you put 3t equal to 3, you can find out what time that happens.

01:28 It happens at time equals 1...

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