An object of mass 200 kg is released from rest from a boat into the water and allowed to sink. While gravity is pulling the object down, a buoyancy force of StartFraction 1 Over 40 EndFraction times the weight of the object is pushing the object up (weight = mg). If we assume that water resistance exerts a force on the object that is proportional to the velocity of the object, with proportionality constant 10 N-sec/m, find the equation of motion of the object. After how many seconds will the velocity of the object be 50 m/sec? Assume that the acceleration due to gravity is 9.81 m divided by sec squared .

= 7.08 s

Explanation:

Given that,

- The mass of the object m = 200 kg

- The buoyancy force F_b = W / 40

- The weight of the object = W = m*g

- Water resistance F_w = k*v

- Where, k = 10 Ns/m

Use Newton's Second law of motion to evaluate the acceleration a.

                                    F_net = m*a

                               W - W/40 - 10*v = m*a

                                a = 39/40*g - 10/m *v

                                a = 39/40 * 9.81 - 10/200 * v

                                a = 9.56 - 0.05*v

- Using the first kinematic equation of motion we have:

                                v_f = v_i + a*t

object was dropped from rest, v_i = 0, Hence:

                                v_f = (9.56 -0.05*v)*t  

                               v_f = 9.56*t / ( 1 + 0.05*t)

Find the time t when v_f = 50 m/s

                                50(1 + 0.05*t) = 9.56*t

                                 50 + 2.5*t = 9.56*t

                                 t = 50 / 7.06

                                     = 7.08 s

massxgravity x heght=weightxheight= pe

4900=490xh

h=10

russian is not my first language, so i'm afraid i can't write out an answer in russian, but hopefully you can find a way to do so. here's my best translation of the problem: you're given a plot of a particle's position in one-dimensional space over time, and you're asked to find either the total distance traveled by the particle after 6 seconds had elapsed, or the total displacement of the particle after 6 seconds.

if you want the total distance traveled, you would first need to determine the particle's velocity as a function of time, then integrate that over the 6-second interval. we know the position function is quadratic with roots at 0 and 6, so we have

where is unknown. because the position is parabolic, there is symmetry about the midpoint at , and we know that

so that the position function is

which means the velocity is

then the total distance traveled is

so the total distance traveled is 20 m.

on the other hand, if you want to find displacement: the particle starts at and ends at , so its displacement is 0.