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Mathematical Methods in the Physical Sciences
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- A spherical dust particle, with a radius a = 0.119 mm and density of 1220 kg/m³, is sedimenting in the quiescent air that has a viscosity of μ =1.81x10-5 kg/m s. The particle experiences the gravitational force, with g = 9.81m/s², as well as a friction force from the viscous air that can be described by F = -6лμаU, where U is the instantaneous velocity of the particle. The motion of the particle is governed by the Newton's second law. At the time t = 0, the dust particle has a downward speed of 0.14 m/s. Use the Euler's method, and a time step of h=0.1s, calculate: When t = 0.1s, the downward speed of the dust particle is m/sarrow_forwardA particle moves through 3-space in such a way that its acceleration is a (t) = 8 sin 2ti +8 cos 2rj+e'k. where t is time measured in seconds. The initial velocity of the particle is v = 2i– 3j+k. At time t = n, find i) the velocity vector of the particle.arrow_forwardConsider the function f(x, y) = y ln x - x ln y - . a Find the directional derivative of f at the point (1, 2) in the direction of the vector v = (-√3, 1). In which direction does f have the maximum rate of change, and b) what is this maximum rate of change?arrow_forward
- A constant force F= (6, 3) (in newtons) acts on a 10-kg mass. Find the position of the mass at t = 10 s if it is located at the origin at t = 0 and has initial velocity vo - (5, -4) (in meters per second). r(10) -arrow_forwardWHICH ANSWER IS CORRECT? Given the curve C parametrized by the vector equation r⃗ (t)=3sin(t)i^+[2−sin(t)+cos(t)]j^−3cos(t)k^,t∈[0;2π] The given curve C has the property that A. Its tangent vector and acceleration vector are always orthogonal. B. The cross product between the tangent vector the acceleration vector always equals −3i^+9j^−3k^. C. The cross product of its acceleration vector and its tangent vector is always parallel to i^+3j^+k^. D. Its vector function is nowhere orthogonal to its tangent vector. E. None of the listed alternatives.arrow_forwardA fish swimming in a horizontal plane has velocity V₁ = (4.00 i + 1.00 j) m/s at a point in the ocean where the position relative to a certain rock is = (14.0 i 3.60 j) m. After the fish swims with constant acceleration for 19.0 s, its velocity is = (17.0 13.00 ĵ) m/s. (a) What are the components of the acceleration of the fish? ax = m/s² ay m/s2 (b) What is the direction of its acceleration with respect to unit vector î? counterclockwise from the +x-axis (c) If the fish maintains constant acceleration, where is it at t = 30.0 s? x = y = m m In what direction is it moving? counterclockwise from the +x-axisarrow_forward
- Suppose that over a certain region of space the electrical potential V is given by the following equation. V(x, y, z) = 2x² - 4xy + xyz (a) Find the rate of change of the potential at P(4, 2, 6) in the direction of the vector v = i + j - k. (b) In which direction does V change most rapidly at P? <20, I (c) What is the maximum rate of change at P?arrow_forwardAt time t=0 a particle at the origin of an xyz-coordinate system has a velocity vector of v0=i+5j−k. The acceleration function of the particle is a(t)=32t^2i+j+(cos2t)k.Find the speed of the particle at time t=1.Round your answer to two decimal places.arrow_forwardThe force on a particle is described by 8x³ - 2 at a point x along the x-axis. Find the work done in moving the particle from the origin to a = 2.arrow_forward
- Trigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage Learning