Question

PLEASE PLEASE PLEASE Explain each part of the solution process. I'm trying to learn this and not just get an answer. I appreciate the help!!

A new speedboat with mass m is traveling at v₀ at some zero time when its engine is shut off. The magnitude of the frictional force f_k between the boat and water is proportional to the speed v of the boat. As an equation, the magnitude of f_k is equal to $\alpha$v, where v is in meters per second, f_k is in newtons, and $\alpha$ is a constant with units of kilograms per second. Assume the boat is moving at speed v at some generic time t. These quantities are known as the base variables.

1. Draw a free-body diagram of the boat when it is slowing down in the water. In this drawing, show the mass, all the acting forces in the proper directions, the direction of motion, and the direction of the acceleration for the speedboat.

2. Find the acceleration a of the speedboat as a function of velocity v and other variables. Is the acceleration constant in time? Carefully explain your answer.

3. Find the velocity v of the speedboat as a function of the base variables, except the acceleration a.

4. Find the position x of the speedboat as a function of the base variables.

5. Find the effective coefficient of friction ($\mu$_k) for the speedboat in the water as a function of the base variables.

6. Sketch your derviced expressions for v and x as functions of time t. Indicate the values for each at t=0 and as t goes to infinity.

Answer 1

The engine is switched off so the only force acting horizontally is the frictional force on the ship. Weightand the up thrust are the two equal force acting against each other in vertical direction enabling the ship to float on water b) This can be found using Newton's second law of motion, F-ma. when the engine is switched OFF the onlyforce acting is the frictional force on the ship and it is equal to cv.So, F ma ma since, fr - av 2 From the above equation acceleration depends on the velocity and the velocity depends on the frictional force.This means more velocity more friction and less velocity less friction. So,the acceleration is not constant with time.

The velocity of the boat can be written from equation of motion. Substitute the values iia αν 2 Since the boat is deceleration, the acceleration should accompany a negative sign. So, 7m