Differential-Equations Velocity and Acceleration Models, guaranteed and verified 100%

1


Velocity and Acceleration Models


A mass near the Earth is under the influence of gravity, which accelerates
the mass toward the Earth at 𝑔 ≈ 9.8𝑚/𝑠𝑒𝑐 2 ≈ 32𝑓𝑡/𝑠𝑒𝑐 2 (assuming we
ignore effects of air resistance). The force on a mass, 𝑚, experiences a force of
gravity given by:



𝐹𝐺 = −𝑚𝑔.


Now let’s consider the impact of the force of air resistance given by:



𝐹𝑅 = −𝑘𝑣 ; 𝑘 > 0.


Note: If an object is falling then 𝑣 is negative, 𝑘 is positive, and
𝐹𝑅 = −𝑘𝑣 is positive.


𝑑𝑣
Newton’s Second Law of Motion: 𝐹 = 𝑚 = −𝑘𝑣 − 𝑚𝑔
𝑑𝑡



𝑑𝑣 𝑘 𝑑𝑣
= −𝑚𝑣 − 𝑔 or = −𝜌𝑣 − 𝑔
𝑑𝑡 𝑑𝑡


𝑘
where 𝜌 = > 0 is called the drag coefficient.
𝑚

, 2


𝑑𝑣
Ex. Let’s solve the separable equation = −𝜌𝑣 − 𝑔.
𝑑𝑡


1 𝑑𝑣 𝑑𝑣
=1 ⟹ = 𝑑𝑡
−𝜌𝑣−𝑔 𝑑𝑡 −𝜌𝑣−𝑔



𝑑𝑣
∫ −𝜌𝑣−𝑔 = ∫ 𝑑𝑡


1
− ln | − 𝜌𝑣 − 𝑔| + 𝑐1 = 𝑡 + 𝑐2
𝜌



1
− ln | − 𝜌𝑣 − 𝑔| = 𝑡 + 𝑐3
𝜌



ln | − 𝜌𝑣 − 𝑔| = −𝜌𝑡 − 𝑐3 𝜌

−𝜌𝑣 − 𝑔 < 0 so |−𝜌𝑣 − 𝑔| = 𝜌𝑣 + 𝑔 and


ln( 𝜌𝑣 + 𝑔) = −𝜌𝑡 − 𝑐3 𝜌


𝜌𝑣 + 𝑔 = 𝑒 −𝜌𝑡−𝑐3 𝜌 = 𝑒 −𝑐3 𝜌 𝑒 −𝜌𝑡


𝜌𝑣 = 𝑒 −𝑐3 𝜌 𝑒 −𝜌𝑡 − 𝑔

1 𝑔
𝑣(𝑡) = 𝜌 (𝑒 −𝑐3 𝜌 𝑒 −𝜌𝑡 ) − 𝜌 .