Question

2. Galileo famously dropped two spheres of different weights (but of the same size) off of the Leaning Tower of Pisa in order to show that Aristotle's theory of gravity (that objects fall at rates proportional to their weights) was wrong. Suppose one sphere had mass m = 1 kg and the other had mass m = 5 kg, and that their velocities followed the equation: mu' = mg - kv with constant of proportionality k = 0.1 kg/s and acceleration g=-10 meters per second squared. (a) Solve for the velocity v(t) of each ball during the fall, assuming that each started with v(0) = 0. (b) How fast was each ball falling after 3 seconds? (c) For each ball, integrate v(t) to get d(t), the distance fallen, assuming (0) = 0. (d) How far had each ball fallen after 3 seconds?

Answer 1

(a)

mu' - kv ku = mg


V k -V=9 m


e tv - e-th -U = e kt 9 772 m


e =e 9


ed e W = e gdt +

$e^{- \frac km t} v = -\frac mk e^{- \frac km t}g + C$
$v(t )= -\frac mk g + C e^{\frac kmt}$

Since ,
$0 = -\frac mkg + C$
$C = \frac mkg$

and
$v(t) = -\frac mkg +\frac mkg e^{\frac km t}$
Replacing the values, we get

.

(b) We need to evaluate at

$v_1(3) = 100 - 100 e^{0.1 \times 3} \approx -34.99$
$v_5(3) = 500 - 500 e^{0.02 \times 3} = -6.18$ .

(c) If $v(t) = -\frac mkg +\frac mkg e^{\frac km t}$ then

$d(t) = \int_0^t v(\tau)\,\text d\tau$
$=\int_0^t-\frac mkg +\frac mkg e^{\frac km \tau}\,\text d\tau$
$=-\frac mkgt +\frac mkg \int_0^t e^{\frac km \tau}\,\text d\tau$
$=-\frac mkgt +\frac mkg \left[ \frac mk e^{\frac km \tau} \right ]_0^t$
$=-\frac mkgt +\frac {m^2}{k^2}g \left( e^{\frac km t}-1 \right )$

Replacing the values,

$d_1(t) =100t -1000 \left( e^{0.1 t}-1 \right )$
$d_5(t) =500t -25000 \left( e^{0.02 t}-1 \right )$

(d) We need to evaluate at ,

$d_1(3) =100\times 3 -1000 \left( e^{0.1 \times 3}-1 \right ) = -49.86$
$d_5(3) =500\times 3 -25000 \left( e^{0.02 \times 3}-1 \right ) = -45.91$ .

They are not the same because of the term which represents, for example, air resistance. But they are very close.