Answer
(a) (i) $v = 9.8 \space m/s$ and $E_k = 0.048 \space J$
(ii) $v = 29 \space m/s$ and $E_k = 0.43 \space J$
Work Step by Step
Free fall: $$r = r_0 + v_0t + \frac 12 at^2$$ $$v = \frac{dr}{dt} = v_0 + at$$ In this case: $v_0 = 0$ $$v = at$$
(i) $$v = at \longrightarrow v = (9.81 \space m/s^2)(1.0 \space s) = 9.81 \space m/s =* 9.8 \space m/s$$ $$E_k = \frac 12 mv^2 = \frac 12 (1.0 \space g)(9.81 \space m/s)^2 = 48 \space g \space m^2/s^2 \times \frac{1 \space kg}{1000 \space g} = 0.048 \space kg \space m^2/s^2 = 0.048 \space J$$
(ii) $$v = at \longrightarrow v = (9.81 \space m/s^2)(3.0 \space s) = 29.43 \space m/s =* 29 \space m/s$$ $$E_k = \frac 12 mv^2 = \frac 12 (1.0 \space g)(29.43 \space m/s)^2 = 430 \space g \space m^2/s^2 \times \frac{1 \space kg}{1000 \space g} = 0.43 \space kg \space m^2/s^2 = 0.43 \space J$$
* Correct number of significant figures.
