Answer
The velocity of the sleigh has a magnitude of 0.105 m/s, and it is moving to the right.
Work Step by Step
We can use conservation of momentum to solve this question. Let $A$ be Jonathan, let $B$ be Jane, and let $C$ be the sleigh. Note that the initial momentum is zero. Let the right direction be the positive direction.
$m_A~v_A~cos(30.0^{\circ})+m_B~v_B~cos(36.9^{\circ})+
m_C~v_C = 0$
$v_C = \frac{-m_A~v_A~cos(30.0^{\circ})-m_B~v_B~cos(36.9^{\circ})}{m_C}$
$v_C = \frac{-(\frac{800~N}{g})(-5.00~m/s)~cos(30.0^{\circ})-(\frac{600~N}{g})(7.00~m/s)~cos(36.9^{\circ})}{\frac{1000~N}{g}}$
$v_C = 0.105~m/s$
The velocity of the sleigh has a magnitude of 0.105 m/s, and it is moving to the right.
