a. A spherical particle of mass m is shot horizontally with initial speed v₀ into a viscous fluid. Use Stokes' law to find an expression for vₓ (t), the horizontal velocity as a function of time. Vertical motion due to gravity can be ignored.

Verified step by step guidance

1. Identify the forces acting on the particle. In this case, the only force acting horizontally is the viscous drag force, which according to Stokes' law is given by F_d = 6\pi\eta r v_x, where \eta is the viscosity of the fluid, r is the radius of the particle, and v_x is the horizontal velocity of the particle.
2. Apply Newton's second law in the horizontal direction. Since the only force is the drag force, the equation becomes m \frac{dv_x}{dt} = -6\pi\eta r v_x, where m is the mass of the particle and \frac{dv_x}{dt} is the acceleration of the particle.
3. Rearrange the equation to separate variables for integration. The equation becomes \frac{dv_x}{v_x} = -\frac{6\pi\eta r}{m} dt.
4. Integrate both sides of the equation. Integrating the left side from the initial velocity v_0 to v_x, and the right side from 0 to t, gives \ln\left(\frac{v_x}{v_0}\right) = -\frac{6\pi\eta r}{m} t.
5. Solve for v_x(t) by exponentiating both sides to eliminate the natural logarithm, resulting in v_x(t) = v_0 e^{-\frac{6\pi\eta r}{m} t}. This is the expression for the horizontal velocity as a function of time.

Key Concepts

Stokes' Law

Stokes' Law describes the force of viscosity acting on a spherical object moving through a viscous fluid. It states that the drag force (F_d) experienced by the object is proportional to its radius (r), the velocity (v), and the viscosity (η) of the fluid, expressed as F_d = 6πηrv. This law is crucial for understanding how the particle's motion is affected by the fluid's resistance.

Terminal Velocity

Terminal velocity is the constant speed that a freely falling object eventually reaches when the resistance of the medium prevents further acceleration. In the context of a particle moving through a viscous fluid, it occurs when the drag force equals the gravitational force acting on the particle, leading to a net force of zero. This concept helps in analyzing the long-term behavior of the particle's velocity.

Newton's Second Law of Motion

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass, expressed as F = ma. In the case of the spherical particle in a viscous fluid, this law is used to relate the forces acting on the particle, including the drag force from the fluid, to its acceleration and ultimately its velocity over time.