A body of mass 100g is dropped from rest toward the earth from a height of 1000m

Question

A body of mass 100g is dropped from rest toward the earth from a height of 1000m. As it falls, air resistance acts upon it, and this resistance (in Newtons) is proportional to the velocity v (in meters per second). Suppose the limiting velocity is 245 m/sec.

(a) Find the velocity and distance fallen at time t secs.

(b) Find the time at which the velocity is one-fifth of the limiting velocity.

NOTE: This problem is for a DIFFERENTIAL EQUATIONS class, NOT A PHYSICS class. Please answer accordingly. Thank you much :)

Explanation / Answer

dv/dt = g - k v for some k

limiting velocity = 245

=> g - k *245 =0

k = g/245

So , taking g =10

equation is

v' = 10 - 10/245v

= 10 - 2/49 v

and initial condition is v(0) = 0

Solution is v = 49/2(10-245e^{-2/49 t + C})

from v(0) = 0 C= 0

So v(t)= 245 - 245 e^{-2/49 t}

y(t) = distance fallen at time t

y(t) = 245t + 245 e^{-2/49 t ) *49/2

b)

v(t) = 245/5 => 245 e^{-2/49 t ) = 245 * 4/5 => e^{-2/49 t ) = 4/5 = > 2/49 t = ln (5/4) => t = 49/2 ln (5/4)

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