Name: Anonymous 2008-03-03 0:33
Hi /sci/, I have a differential equation question for you.. math stuff.. someone please give me a hint and help me out this is bugging the crap out of me.
Sorry for the HW question, but this is purely to help me understand this... the book doesn't explain the direct connection between the limiting velocity and air resistance.
5. 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 245m/s.
a. Find the velocity and disnce fallen at time t secs.
b. Find the time at which the velocity is one-fifth of the limiting velocity.
Answers:
a. v=245[1-e^(-t/25)] , x=245[t+25(e^(-t/25)-1)]
b. t=5.58
I set up my equation as
100dv/dt = 981 - kv
Differentiated by seperation,
981-kv=e^(kt)*e^(kC1) , used the initial value v(0)=0
C1=ln(981)/k
Plugged that back in...
v=(981/k)*[1-e^(kt)]
How do I get from here to the first answer?? I don't get how the limiting velocity relates to my v equation.. I think it's the same thing as the limit of v as it approaches infinity, but that makes my equation make no sense. Someone please help!
Sorry for the HW question, but this is purely to help me understand this... the book doesn't explain the direct connection between the limiting velocity and air resistance.
5. 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 245m/s.
a. Find the velocity and disnce fallen at time t secs.
b. Find the time at which the velocity is one-fifth of the limiting velocity.
Answers:
a. v=245[1-e^(-t/25)] , x=245[t+25(e^(-t/25)-1)]
b. t=5.58
I set up my equation as
100dv/dt = 981 - kv
Differentiated by seperation,
981-kv=e^(kt)*e^(kC1) , used the initial value v(0)=0
C1=ln(981)/k
Plugged that back in...
v=(981/k)*[1-e^(kt)]
How do I get from here to the first answer?? I don't get how the limiting velocity relates to my v equation.. I think it's the same thing as the limit of v as it approaches infinity, but that makes my equation make no sense. Someone please help!