Donald Johnson

2021-12-29

A steel ball of mass 4-kg is dropped from rest from the top of a building. If the air resistance is 0.012v and the ball hits the ground after 2.1 seconds, how tall is the building? Answer in four decimal places.

Piosellisf

Beginner2021-12-30Added 40 answers

Mass of the ball $m=4kg$

Dropped from nest do initial velocity is zero is$v=0\text{}\text{when}\text{}t=0$

Given that air resistence$=0.012v$

The equation of motion is

$m\frac{dv}{dt}=g-0.012v$

or$\frac{dv}{dt}=\frac{g-0.012v}{m}$

$=\frac{g-0.012v}{4}$

$\frac{dv}{g-0.012v}=\frac{dt}{4}$

or$\frac{d(g-0.012v)}{-0.012(g-0.012v)}=\frac{dt}{4}$

or$\frac{d(g-0.012v)}{g-0.012v}=\frac{dt}{4}$

$=-0.003dt$

Integrating$\int \frac{d(g-0.012v)}{g-0.012v}=-0.003\int dt+c$

or$\mathrm{ln}(g-0.012v)=-0.003t+C$

Initially$v=0,\text{when}\text{}t=0$

$\therefore PSK\mathrm{ln}\left(g\right)=c$

$\therefore \mathrm{ln}(g-0.012v)=-0.003t+\mathrm{ln}g$

$\mathrm{ln}\left(\frac{g-0.012v}{g}\right)=-0.003t$

$\frac{g-0.012v}{g}={e}^{-0.003t}$

$1-\frac{0.012v}{g}={e}^{-0.003t}$

$1-{e}^{-0.003t}=\frac{0.012}{g}v$

$v=\frac{g}{0.012}(1-{e}^{-0.003t})$

$\frac{dx}{dt}=\frac{g}{0.012}(1-{e}^{-0.003t})$

$dx=\frac{g}{0.012}(1-{e}^{-0.003t})dt$

Integrating,

Dropped from nest do initial velocity is zero is

Given that air resistence

The equation of motion is

or

or

or

Integrating

or

Initially

Integrating,

raefx88y

Beginner2021-12-31Added 26 answers

Solution:

Given, mass of the ball,$m=4kg$

air resistance,$=0.015v$

$t=3.4$ seconds

Now, speed at the ball at the bottom

$v=u+>$

$=0+9.8\times 3.4$

$=33.32\frac{m}{s}$

Resistance$=0.015v$

$=0.015\times \left(33.32\right)$

$=0.5\frac{m}{{s}^{2}}$

$h=u+{y}_{2}g+2$
NSk
$h=0+\frac{1}{2}(9.8-0.5)\times {3.4}^{2}$

$h=53.754m$

Height of the building is 53.754m

Given, mass of the ball,

air resistance,

Now, speed at the ball at the bottom

Resistance

Height of the building is 53.754m

karton

Expert2022-01-09Added 613 answers

Mass of the body m=4 kg, time taken to hit the ground t=3.7s and air resistance f=0.013v. Since the ball was dropped, the initial velocity u=0.

Force equation for the ball, F=ma=mg-f

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