- #1

ma18

- 93

- 1

## Homework Statement

This is a standard projectile motion problem, the mass is m, the drag is

**F**_r = - alpha *

**v,**where alpha is a positive retarding coefficient. The origin is the ground and at time t = 0 the horizontal and vertical velocities are positive

a) Write down Newton's second law for the horizontal component of the velocity, v_x (t). Solve for v_x (t) with given initial conditions and use the characteristic time, tau = m/alpha in your answer. Give

**physical**reasoning as to why m increases with tau and why alpha decreases with tau.

b) Integrate v_x (t) from a to determine x(t) with given initial conditions. What is x(t) with the limit t goes to infinity? What would be the horizontal position at t = tau?

c) Convert the DE from a into an equation for v_x (x). Solve for v_x (x) with the given initial conditions and use tau in your answer.

## Homework Equations

F = ma

## The Attempt at a Solution

a)

F_x = m*v'_x = - alpha*v_x

v'_x/v_x = d/dt *ln(v_x) = - alpha * v_x

ln v_x = - alpha/m t + ln v_x0

v_x (t) = v_x0 * e^(-t/tau)

b) I am stuck here, I integrate to get

integral v_x (t) = x(t) = v_x0 *e^(1-t/tau) / (1-t/tau) * A (where A is an arbitrary constant of integration)

But solving for A with the condition that x(t=0) = 0 I get A = 0

c) I am stuck here too,

m dv_x/dx * dx/dt = - alpha * v(t)

I don't know how the convert the second v(t) into a v(x)

Any help would be much appreciated.