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.

Question

Accepted Answer

Hey, everyone. So this problem is working with drag forces. Let's see what they're asking us at an initial speed of V. Sub B B gun fires a metallic ball of mass M sub B and radius R sub be horizontally into a barrel containing sunflower oil of viscosity to find an algebraic expression for the time varying horizontal velocity by applying Stokes lock to the ball. And they tell us to neglect the gravitational force. So we can call that Stokes law is given by the drag force is equal to six pi A to R V. And from Newton's 2nd law, we know that the drag force which is negative because it's acting in the direction opposite to the motion of the ball is equal to mass times acceleration. So acceleration we can recall is the derivative of velocity with respect to time. We'll write that as A equals T V D T. And so when we substitute that in and apply Stokes law, we'll get negative six pi A to R V equals mm TV T T one V and D V on the same side of the equation. And so we'll rearrange this to be T V over B equals negative six hi data are over em multiplied by D T when we integrate those from V I to be of T of D V over V. And then on the other side of the equation, we will keep the constants out and integrate D T From T equals zero to T. So the integral of D V over V we can recall is the L N of V of tea over the eye. And that is equal to six pi data are over M multiplied by T. And so we're gonna take both sides this key to the power of and so even the power of L N cancels. So we have V sub T over V I equals E to the six pi A to our over em multiplied by T. And then finally, to get B of T by itself, it will multiply V I or the initial two, the other side. And we can recall from the problem but they said the initial was Visa B and then the mass and the radius are also our sub B and M sub B respectively. So we'll rewrite that V sub B multiplied by E to the negative six pi A to our sub B over M sub be multiplied by T. And that is our algebraic equation or philosophy. And so when we go up to our potential answers, we see that, that aligns with answer choice. See, so C is the correct answer for this problem. That's all we have for this one. We'll see you in the next video.

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 Solution

Key Concepts

Stokes' Law

Terminal Velocity

Newton's Second Law of Motion