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
Trending nowThis is a popular solution!
Chapter 12 Solutions
Multivariable Calculus
- Velocity of a Boat The boater in Exercise 63 wants to arrive at a point on the north shore of the river directly opposite the starting point In what direction should the boat be headed?arrow_forwardVelocitySuppose 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_forwardInclined Ramp In Exercises 8992, a force of F pounds is required to pull an object weighing W pounds up a ramp inclined at degrees from the horizontal. Find when F=5000 pounds and W=15,000 pounds.arrow_forward
- Finding the Actual Speed and Direction of an Aircraft ABoeing 747 jumbo jet maintains a constant airspeedof 550 miles per hour (mph) headed due north. Thejet stream is 100 mph in the northeasterly direction.(a) Express the velocity va of the 747 relative to the air andthe velocity vw of the jet stream in terms of i and j.(b) Find the velocity of the 747 relative to the ground.(c) Find the actual speed and direction of the 747 relative tothe ground.arrow_forwardThe position vector r describes the path of an object moving in space. Find the velocity v(t), speed s(t), and acceleration a(t), of the object. r(t) = ti + t²j+ 22k 2 v(t) = s(t) = a(t) = Need Help? eBook Watch Itarrow_forwardFind a vector equation and parametric equations for the line. (Use the parameter t.) The line through the point (6 -5.) and parallel to the vector (1.3.-3) r(t) %3D (x(t), v(t), zít)arrow_forward
- Another 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_forward(a) Find a direction vector for the first line, which is given in parametric form. (b) Find parametric equations for the second line in terms of the parameter t. (c) Show that the two lines intersect at a single point by finding the values of s and t that result in the same point. Then find the point of intersection.arrow_forward5: A potato cannon launches a potato straight up from a height of 5 feet with an initial velocity of 128 ft/sec. The horizontal distance from the cannon launch and the vertical height from the ground, in feet, are modeled by a set of parametric equations. . Write a set of parametric equations that will model the height of the potato as a function of time, t. . Find the average rate of change in horizontal position over the first two seconds. Show your work and units. What is the average rate of change in vertical motion over the first two seconds? Show your work and units.arrow_forward
- True Velocity of a Jet A jet is flying through a wind that is blowing with a speed of 55 mi/h in the direction N 30° E (see the figure).The jet has a speed of 765 mi/h relative to the air ,and the pilot heads the jet in the direction N 45° E. (a) Express the velocity of the wind as a vector in component form. (b) Express the velocity of the jet relative to the air as a vector in component form. (c) Find the true velocity of the jet as a vector. (d) Find the true speed and direction of the jet.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. Position Vector r(t) = √ti + 5tj + 2t2k Time t = 4arrow_forwardBill's train leaves at 8:06 AM and accelerates at a rate of 4 meters per second per second. Bill, who can run 6 meters per second, arrives at the train station 6 seconds after the train has left. (a) Find parametric equations that model the motions of the train and Bill as a function of time, (Hint: The position x at time t of an object having acceleration a is x = at.) (b) Determine algebraically whether Bill will catch the train. If so, when? (c) Simulate the motion of both by simultaneously graphing the equations found in part (a). (a) Let y, = 2 be the path of Bill and y, = 4 be the path of the train. What are the parametric equations describing Bill's motion? O x, = 22, y, = 2 O x, = 6t, y, = 2 O x1 = 6(t + 6), y, = 2 O X1 " 6(t - 6). y, = 2 What are the parametric equations describing the train's motion? O x2 = 41, y2 = 4 O x2 = 21, y2 = 4 O x = 21, y2 = 4 O x, = 3, y2 = 4 IIarrow_forward
- Algebra and Trigonometry (MindTap Course List)AlgebraISBN:9781305071742Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage LearningAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageTrigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage Learning