Answer:
To sketch the velocity field, we can plot a set of velocity vectors at various points in the domain. Here, we will plot the vectors at a grid of points in the xy-plane.
First, let's plot the vector field using Python:
import numpy as np
import matplotlib.pyplot as plt
# Define the velocity field functions
def u_func(x, y):
return y**2 - x**2
def v_func(x, y):
return 2*x*y
# Define the grid of points
x = np.linspace(-3, 3, 20)
y = np.linspace(-3, 3, 20)
X, Y = np.meshgrid(x, y)
# Compute the velocity components at each point in the grid
U = u_func(X, Y)
V = v_func(X, Y)
# Plot the vector field
fig, ax = plt.subplots(figsize=(6, 6))
ax.quiver(X, Y, U, V)
ax.set_xlabel('x')
ax.set_ylabel('y')
ax.set_xlim(-3, 3)
ax.set_ylim(-3, 3)
plt.show()
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To find the velocity and acceleration components at points (2,2) and (2,-2), we first need to evaluate the velocity field functions at these points:
At (2,2):u = y^2 - x^2 = 2^2 - 2^2 = 0
v = 2xy = 2*2*2 = 8
So the velocity vector at (2,2) is (0, 8).
To find the acceleration components, we need to compute the partial derivatives of the velocity field functions with respect to x and y:
a_x = ∂u/∂x = -2x
a_y = ∂u/∂y = 2y
So at (2,2), the acceleration vector is (-4, 4).
At (2,-2):u = y^2 - x^2 = (-2)^2 - 2^2 = -4
v = 2xy = 2*2*(-2) = -8
So the velocity vector at (2,-2) is (-4, -8).
To find the acceleration components, we again need to compute the partial derivatives of the velocity field functions:a_x = ∂u/∂x = -2x
a_y = ∂u/∂y = 2y
So at (2,-2), the acceleration vector is (-4, -4).
Step-by-step explanation: