Finding Velocity and Acceleration Along a Plane Curve In Exercises 3-10, 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
Position Vector Point
Want to see the full answer?
Check out a sample textbook solutionChapter 12 Solutions
Multivariable Calculus
- VelocitySuppose that in Exercise 55 the current is flowing at 1.2 mi/hr due south. In what direction should the swimmer head in order to arrive at a landing point due east of his starting point? VelocityA river flows due south at 3mi/h. A swimmer attempting to cross the river heads due east swimming at 2mi/h relative to the water. Find the true velocity of the swimmer as a vector.arrow_forwardThe position vector r describes the path of an object moving in space. (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 valueof t.arrow_forwardthe position vector r describes the path of an object moving in space. (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 value of t. r(t) = ⟨t, −tan t, e^t⟩ t = 0arrow_forward
- A baseball leaves the hand of a pitcher 6 vertical feet above and 60 horizontal feet from homeplate. Assume the coordinate axes are such that the origin is at ground level directly below the point ofrelease. A simple model to describe the curve of a baseball assumes the spin of the ball producesa constant sideways acceleration (in the y-direction) of c ft/s^2. Suppose a pitcher throws a curve ballwith c = 8 ft/s^2. How far does the ball move in the y-direction by the time it reaches home plate,assuming an initial velocity of < 130,0,−3 > ft/s? Does the ball curve more in the first half of its trip to the plate or in the second half? Justify your answer.arrow_forwardFinding Curvature In Exercises 23-28, find the curvature of the plane curve at the given value of the parameter. r(t)=(t, sent) , t=pi/2arrow_forwardThe position vector r describes the path of an object moving in space. (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 value of t. Position Vector r(t) = ⟨t, −tan t, et⟩ Time t=0arrow_forward
- Sketch the plane curve represented by the vector valued function and give the orientation of the curve. r(t) = (t2 + t)i + (t2 − t)jarrow_forwardAnalyzing motion Consider the position vector of the following moving objects.a. Find the normal and tangential components of the acceleration.b. Graph the trajectory and sketch the normal and tangential components of the acceleration at two points on the trajectory. Show that their sum gives the total acceleration. r(t) = 2 cos t i + 2 sin t j, for 0 ≤ t ≤ 2πarrow_forwarda. Find the length of the curve. b. Find the tangent vector T at point (1, 2\sqrt{2}, 3). c. Find the normal vector N at point (1, 2\sqrt{2}, 3). d. Find the binormal vector B at point (1, 2\sqrt{2}, 3). e. Find the velocity and acceleration of moving point P with position r(t). f. Find the curvature at general point.arrow_forward
- Curve In Exercise 56, sketch the plane curve and find its length over the given interval. 56. r(t) = t 2i + 2tk, [0, 3]arrow_forwardAnother crosswind flight A model airplane is flying horizontally due east at 10 mi>hr when it encounters a horizontal crosswind blowing south at 5 mi/hr and an updraft blowing vertically upward at 5 mi/hr.a. Find the position vector that represents the velocity of the plane relative to the ground.b. Find the speed of the plane relative to the ground.arrow_forwardThe position vector r(t) = ti + 5tj + 3tk describes the path of an object moving in space. (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 value of t=1.arrow_forward
- Algebra and Trigonometry (MindTap Course List)AlgebraISBN:9781305071742Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage LearningTrigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage LearningAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage