- #1

CAF123

Gold Member

- 2,948

- 88

## Homework Statement

A particle of mass ##m## is projected from ##x(0) = x_o## in the vertical direction, with an initial velocity, ##\dot{x}(0) = v_o##. It is subject to gravity and linear drag, mk|v|, against the motion.

1) Show that the body follows ##v(t)## such that:$$v(t) = -g/k + (v_o + g/k)e^{-kt}$$

2) Use ##\ddot{x} = v dv/dx ## to find a soln to the motion subject to the initial condition. Show the equivalence between the result attained and the one shown in 1).

## Homework Equations

Separable Diff Eqns, Newton 2nd

## The Attempt at a Solution

1) is fine. I don't really know how to show the equivalence, but I have tried two different ways, where one way I get nothing near equivalence and the other I recover the exp term above, but not the -g/k in front.

Method 1). After solving ##\ddot{x} = v dv/dx, ##I get ##x = x(v)##. Then by the chain rule, ##dx/dt = dx/dv\,dv/dt##. So I differentiated my soln in 2) wrt v and then multiplied this by the derivative of v wrt t in 1). This gives me nothing near equivalence, although I am not sure why.

Method 2). Put v = v(t) in 1) into my x(v). This gives x(t). Then simply differentiate wrt t. I get the second term in 1) (exp term) but not the -g/k.

Both methods seem valid, (are they?) but I don't get the result. I could recheck my algebra again, but both methods yield a (g +kv)^3 and that is not present in 1) and it doesn't cancel.

Many thanks.