# Part (a):
#
m = 0.25
g = 9.8
k = 0.2
x0 = 30

x = function(t){
  x0 - (g*m*t)/k + ( (g*m^2)/(k^2) ) * ( 1 - exp(-k*t/m) )
}
v = function(t){
  ((g*m)/k)*( exp(-k*t/m) - 1 )
}

ts = seq( 0, 5, length.out=100 )
xs = x(ts)

plot(ts, xs, type='l')
grid()

library(stats)
res = uniroot( x, c(3, 4) )

t_g = res$root
v_g = v(t_g)
print( sprintf('t_g= %f v(t_g)= %f', t_g, v_g) )

# Part (b):
#
x0_lt = 10 # velocity on impact is less than 10 m/s
x0_gt = 30 # velocity on impact is greater than 10 m/s
while( abs(x0_gt-x0_lt)>0.01 ){
    x0 = 0.5 * ( x0_lt + x0_gt )

    # What time do we hit the ground:
    res = uniroot( x, c(0, 4) )

    # What is our impact velocity when we hit:
    v_impact = v(res$root)
    if( abs(v_impact)>10 ){
        x0_gt = x0
    }else{
        x0_lt = x0
    }
    print( c(x0_lt, x0_gt) )
}

# Part (c):
#
x0 = 30 # set x0 back to 30
k_gt = 0.2 # velocity on impact is greater than 10 m/s
k_lt = 0.5 # velocity on impact is less than 10 m/s
while( abs(k_gt-k_lt)>0.001 ){
    k = 0.5 * ( k_lt + k_gt )

    # What time do we hit the ground:
    res = uniroot( x, c(0, 10) )

    # What is our impact velocity when we hit:
    v_impact = v(res$root)
    if( abs(v_impact)>10 ){
        k_gt = k
    }else{
        k_lt = k
    }
    print( c(k_gt, k_lt) )
}