Predicting Particle Movement with Air Resistance: A Simple Example"

[Georgepowell]

I have done M1, M2, C1 C2 C3 & C4 (A level maths modules, M meaning mechanics and C meaning core/pure). In M2 we learned how to predict how a particle moves under gravity when projected at an angle, and we always ignored air resistance. In C4 we have learned how to integrate a whole range of functions. This is not homework or school work

I wondered how to predict the movement of a particle if the air resistance WAS taken into account. So I tried to figure it out and I just need to know if I have done it right. Here is the [very simple] example:

A particle of mass one kg is initially traveling at 10m/s, air resistance is equal to a tenth of its current velocity. Find its velocity in terms of time.

I wrote:

F = ma
v/10 = a
a = v/10

acceleration is also dv/dt so:

dv/dt = v/10
1/v dv = 1/10 dt
integrate both sides you get:
ln(v) = t/10 + lnA
v = Ae^(t/10)

t = 0, v = 10

10 = Ae^0

so final answer is:

v = 10e^(t/10)

is this right? is there another way to do it?

 

[Astronuc]

That's almost right.

a = -v/10, meaning that acceleration is negative (deceleration), i.e. v is decreasing.

There is an initial velocity v_o = 10 m/s, at t = 0.

Make sure the limits of integration are correct.

 

[Georgepowell]

That's almost right.

a = -v/10, meaning that acceleration is negative (deceleration), i.e. v is decreasing.

There is an initial velocity v_o = 10 m/s, at t = 0.

Make sure the limits of integration are correct.

Thanks, although haven't I already taken the initial velocity into account?

t = 0, v = 10

10 = Ae^0

so final answer is:

v = 10e^(t/10)