Concept explainers
(a) Sketch the
(b) If parametric equations of the flow lines are
(c) If a particle starts at the origin in the velocity field given by F, find an equation of the path it follows.
Want to see the full answer?
Check out a sample textbook solutionChapter 16 Solutions
CALCULUS WEBASSIGN <CUSTOM>
- Example :- Find the velocity, speed, and acceleration of particle whose motion in space is given by the position vector FH=(2 0st)i+(2 sint)j+(5 cos't) - Find the velocity vector V ()?arrow_forwardThe position vector r describes the path of an object moving in the xy-plane. Position Vector Point r(t) = t'i + tj (4, 2) (a) Find the velocity vector, speed, and acceleration vector of the object. v(t) = s(t) a(t) = (b) Evaluate the velocity vector and acceleration vector of the object at the given point. v(2) = a(2) =arrow_forwardIf r(t) = cos(7t)i + sin(7t)j – 3tk, compute the tangential and normal components of the acceleration vector. COS Tangential component ar(t) Normal component an(t) =arrow_forward
- An object moves with an acceleration vector: a(t) = - 3 Cos(t) i - 3 Sin(t) j + 2k If initially the object starts from the point (3,0,0), with a speed v(0)=3j , find: The position r(t) of the object for all time t and the normal and tangential components of acceleration.arrow_forwardIf r(t) = cos(2t)i + sin(2t)j – 4tk, compute the tangential and normal components of the acceleration vector. Tangential component ar(t) = Normal component a(t) =arrow_forwardThe acceleration vector for the spacecraft Dolphin 163 is given by a(t) = (-2 cos(t), 0,-2 sin(t)). It is also known that the velocity and position att = 0 are (0) = (0, v5, 2) and F(0) = (3, 0,0 ). Assume distances are measured in kilometers (km) and time is measured in seconds (s). (a) Find the position function F(t) for the spacecraft. (b) Find the function for the speed of the spacecraft and the speed when t = 0. (c) Compute the curvature of the trajectory when t 0. (d) At time t = A seconds the spacecraft launches a probe in a direction opposite of N, the unit normal vector to 7. If the probe travels along a straight line in the direction it was launched from the spacecraft for 5 km and then stops, what is its resting coordinate?arrow_forward
- At time t = 0, a particle is located at the point (1, 2, 3). (Vector Functions) It travels in a straight line to the point (4, 1, 4), has speed 2 at (1, 2,3) and constant acceleration 3i – j+k. Find equation for the position vector r(t) of the particle at time t.arrow_forwardSketch the curve whose vector equation is Solution r(t) = 6 cos(t) i + 6 sin(t) j + 3tk. The parametric equations for this curve are X = I y = 6 sin(t), z = Since x² + y² = + 36. sin²(t) = The point (x, y, z) lies directly above the point (x, y, 0), which moves counterclockwise around the circle x² + y2 = in the xy-plane. (The projection of the curve onto the xy-plane has vector equation r(t) = (6 cos(t), 6 sin(t), 0). See this example.) Since z = 3t, the curve spirals upward around the cylinder as t increases. The curve, shown in the figure below, is called a helix. ZA (6, 0, 0) (0, 6, 37) I the curve must lie on the circular cylinder x² + y² =arrow_forwardEXO: Find the phase flows of the systems i-Sinyarrow_forward
- At time t=0, a particle is located at the point (3,9,4). It travels in a straight line to the point (7,8,6), has speed 6 at (3,9,4) and constant acceleration 4i-j+2k. Find an equation for the position vector r(t) of the particle at time t -O+¹+* The equation for the position vector r(t) of the particle at time t is r(t) = (Type exact answers, using radicals as needed.)arrow_forward(f) List all the equilibrium solution in the direction field shown above in the chart.arrow_forwardThe acceleration vector for the spacecraft Dolphin 163 is given by d(t) = (-2 cos(t), 0,–2 sin(t)). It is also known that the velocity and position at t = 0 are ü(0) = (0, V5, 2) and r(0) = (3, 0,0 ). Assume distances are measured in kilometers (km) and time is measured in seconds (s). (a) Find the position function F(t) for the spacecraft. (b) Find the function for the speed of the spacecraft and the speed when t = 0. (c) Compute the curvature of the trajectory when t = 0. (d) At time t = T seconds the spacecraft launches a probe in a direction opposite of N, the unit normal vector to r. If the probe travels along a straight line in the direction it was launched from the spacecraft for 5 km and then stops, what is its resting coordinate?arrow_forward
- Calculus: Early TranscendentalsCalculusISBN:9781285741550Author:James StewartPublisher:Cengage LearningThomas' Calculus (14th Edition)CalculusISBN:9780134438986Author:Joel R. Hass, Christopher E. Heil, Maurice D. WeirPublisher:PEARSONCalculus: Early Transcendentals (3rd Edition)CalculusISBN:9780134763644Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric SchulzPublisher:PEARSON
- Calculus: Early TranscendentalsCalculusISBN:9781319050740Author:Jon Rogawski, Colin Adams, Robert FranzosaPublisher:W. H. FreemanCalculus: Early Transcendental FunctionsCalculusISBN:9781337552516Author:Ron Larson, Bruce H. EdwardsPublisher:Cengage Learning