Answer
The flea goes up to a height of 5.15 cm
Work Step by Step
We can find the velocity $v$ after the flea accelerates through the first 0.50 mm:
$v^2= v_0^2+2ay = 0 + 2ay$
$v = \sqrt{2ay} = \sqrt{(2)(1000~m/s^2)(0.50\times 10^{-3}~m)}$
$v = 1.0~m/s$
We then find the distance the flea goes up after the initial period of acceleration;
$y = \frac{0-v^2}{2a} = \frac{-(1.0~m/s)^2}{(2)(-9.80~m/s^2)}$
$y = 0.0510~m$
The total height is $0.0510$ m plus the initial 0.50 mm while the flea was accelerating upward. This makes the total height 0.0515 meters which is 5.15 cm.
