Answer
(a) The speed at the bottom of the hill is 8.16 m/s.
(b) The internal energy generated when crossing the rough patch was 766 J.
Work Step by Step
$K_2 + U_2 = K_1 + U_1 + W_f$
$\frac{1}{2}mv_2^2+0 = \frac{1}{2}mv_1^2+ mgh - mg~\mu_k~d$
$v_2^2 = v_1^2+ 2gh - 2g~\mu_k~d$
$v_2 = \sqrt{v_1^2+ 2gh - 2g~\mu_k~d}$
$v_2 = \sqrt{(6.50~m/s)^2+ (2)(9.80~m/s^2)(2.50~m) - (2)(9.80~m/s^2)(0.300)(4.20~m)}$
$v_2 = 8.16~m/s$
The speed at the bottom of the hill is 8.16 m/s.
(b) The internal energy is equal in magnitude to the work done by friction when crossing the rough patch.
$W_f = -mg~\mu_k~d$
$W_f = -(62.0~kg)(9.80~m/s^2)(0.300)(4.20~m)$
$W_f = -766~J$
The internal energy generated when crossing the rough patch was 766 J.
