# Integration of dynamics problem

1. May 30, 2010

### fysiikka111

1. The problem statement, all variables and given/known data
A package of mass m is released from an aircraft flying horizontally with a velocity v0. In addition to gravity, a drag force acts on the package given by F=-$$\lambda$$v. Find the i and j components of the velocity as a function of the time (t) from when the package was released.

2. Relevant equations
F=ma

3. The attempt at a solution
Acceleration in x-direction:
F=ma(x)
-$$\lambda$$v(x)=ma(x)
a(x)=-$$\lambda$$v(x)/m
dv(x)/dt = -$$\lambda$$v(x)/m
The solution method then integrates, and gets:
v(x) = v0exp(-$$\lambda$$t/m)
I don't know how to do the integration.
Thanks.

2. May 30, 2010

### Cyosis

They move all v-s to one side and t-s to the other side.
$$\frac{dv}{v}=-\frac{\lambda}{m} dt$$

You can now easily integrate this equation and solve for v.

3. May 30, 2010

### fysiikka111

So,
dv/v = -$$\lambda$$dt/m
ln v = -$$\lambda$$t/m
v = exp(-$$\lambda$$t/m)

But in the answer, where does the extra v0 come from that is before the exponent?

4. May 30, 2010

### Cyosis

You forgot the integration constant in combination with $v(t=0)=v_0$.