Evaluate the line integral of the velocity u= √y, v=x-y over the following paths (a) x=t, y=t (b) x=12, y=t where 0≤1. (c) x=t, y = 12 (d) x=t, y = t³
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- (d) Given a velocity graph as shown in Figure Q.2(d). Construct the a – t and s – t graphs. ERSIT v (m/s) v = 0.6t3/ t(s) 15 25 Figure Q.2(d)When a valve is opened, a certain fluid flows through the choke duct or valve (see figure), according to the relationship: V= V (1 + x/L) i Determine a) If the flow is stationary or transient. b) The acceleration (ax) of the fluid applying Euler's approach. c) The position of the particle as a function of time at x = 0 and t = 0. d) Determine the acceleration of the particle as a function of time.The system, dx dt is a potential flow. = -12 + 16x + 3x² - 4x³ True False
- A flow is uniform if the parameters describing the flow vary with distance along the flow path. TRUE FALSEy x = r cos 0 V = Or y = r sine r = √x² + y² χ Flow in "solid body rotation" acts like a solid spinning around an axis. The streamlines are circular, the velocity is purely tangential, and the velocity magnitude is V = r, where is the angular velocity (positive counter-clockwise) and r is the radius. (a) Express the velocity vector V as a function of x and y. (b) Calculate the curl of the velocity vector V × V, indicating clearly the direction of the resulting vector.3.1. The velocity at a point in a fluid for a one-dimensional flow may be given in the Eulerian coordinates by u == AxBt. Show that x = f(x, t) in the Lagrange coordinates can be obtained from the Eulerian system. The in- itial position of the fluid particle is designated by x) and the initial time to = 0 may be assumed.
- 1. Which of the following is a linear equation? (Select all that apply) (а) 12 + у — 19 (c) sin x + 9y = -4 (e) x² – y? = 1 -3x + 4y 2x (d) (b) х 3 —у + 2 Зу - (f) VT = y 10 5 2. Give an example of the following of a linear combination in t, u, v, w.(3a)The first time derivative of vector A cross vector B is equal to the first time derivative of vector B cross vector A. The first time derivative of vector A cross vector B is equal to the first time derivative of vector B cross vector A. True False (b) Suppose you're interested in the first time-derivative of velocity vector v(t) = (5t2)i + (2t)j. Which of the following expressions represents the first time-derivative of two times v(t)? Suppose you're interested in the first time-derivative of velocity vector v(t) = (5t2)i + (2t)j. Which of the following expressions represents the first time-derivative of two times v(t)? (10t)i + (2)j (20t)i + (4)j (10)i (5/3)(t3)i + (t2)jHome Work (steady continuity equation at a point for incompressible fluid flow: 1- The x component of velocity in a steady, incompressible flow field in the xy plane is u= (A /x), where A-2m s, and x is measured in meters. Find the simplest y component of velocity for this flow field. 2- The velocity components for an incompressible steady flow field are u= (A x* +z) and v=B (xy + yz). Determine the z component of velocity for steady flow. 3- The x component of velocity for a flow field is given as u = Ax²y2 where A = 0.3 ms and x and y are in meters. Determine the y component of velocity for a steady incompressible flow. Assume incompressible steady two dimension flow
- 1. Suppose we have a robot moving in a 2D plane with dynamics x(t+1) = x(t) + v(t), v(t+1)=0.5v(t) + a(t), where x(t), v(t) and a(t) denote its location, velocity and acceleration, respectively. Let the robot be initially at rest at a starting location (1) = [3] 1] it comes to a full stop. The goal is to design an optimization problem that moves the robot to its desired location while minimizing the total acceleration ₁ ||a(t)||2, satisfying the above dynamics, and with an additional constraint that the velocity magnitude (2-norm) is no larger than 0.15. It has to reach a final location x (T) = at T = 50, at which (a) Formulate the above optimization problem. Is it convex? What category does it fall into? Solve it using CVX. Plot the trajectory of the robot in the 2D plane, plot ||v(t)||2 and ||a(t)||2 versus t. (b) Now suppose we would like the robot to avoid a certain obstacle represented by a region in the 2D plane. Specifically, the path of the robot should not go through a circle…1) Particle Motion and Coordinate Systems y y = 0.001x² %3D 100 m The jet pictured above is following the given curve. All of the question below are related to the moment pictured a) If the jet's altitude is increasing at a rate of 40 m/s, find its velocity vector (in Cartesian) b) Find the speed of the jet c) In addition, if the rate of change of the jet's vertical speed is increasing at 15 m/,2, find the acceleration vector d) Find the tangential component of the acceleration vector, v e) Find the normal component of the acceleration vector f) Find the radius of curvature of the jet's path using the information from parts b) and e)#4 1.11 For a small particle of styrofoam (1 lbm/ft) (spherical, with diameter d = 0.3 mm) falling in standard air at speed V, the drag is given by FD-3mVd, where is the air viscosity. Find the maximum speed starting from rest, and the time it takes to reach 95 percent of this speed. Plot the speed as a function of time. s) Answer: (Vmax=0.0435",t=0.0133 S