Question

After sledding down a hill, a boy and girl are sitting at rest. The combined mass of the kids and the sled is 210 kg. The 70 kg boy jumps north at 6.5 m/s and the 50 kg girl jumps to the south at 8.1 m/s. How fast and in what direction will the sled be traveling?

Answer 1

Let $m_b = 70kg, m_g = 50kg$ be the masses of the boy and the girl respectively. The mass of the sled is then:

$$m_s = 210kg-m_b-m_g = 90kg$$

Let also $v_b = 6.5m/s, v_g = 8.1m/s$ be the speeds of the boy and the girl respectively. They give the following momentums to the sled:

$$p_b = m_bv_b = 70kg\cdot 6.5m/s =455\space kg\cdot m/s\\ p_g = m_gv_g = 50kg\cdot 8.1m/s =405\space kg\cdot m/s$$

Since the momentums are applied in the opposite directions, the resultant momentum given to the sled is:

$$p_s = p_b-p_g = 50\space kg\cdot m/s$$

The speed of the sled is then:

$$v_s = \dfrac{p_s}{m_s} = \dfrac{50\space kg\cdot m/s}{90kg} \approx 0.56m/s$$

Since the boy's momentum was greater then the girl's one, the resultant velocity of the sled is directed to the south (since the boy jumped to the north).

Answer. 0.56 m/s to the north.