A 50 kg girl who is running at 1.5 m/s jumps on a stationary 8.5 kg sled on a frozen lake. How fast does the sled with the girl then move?

To find the velocity of the sled with the girl after she jumps onto it, we can use the principle of conservation of momentum. According to this principle, the total momentum before the jump should be equal to the total momentum after the jump. The formula for momentum is: Momentum (p) = Mass (m) * Velocity (v) Let's denote the initial velocity of the girl as v_girl_initial and the initial velocity of the sled as v_sled_initial, and the final velocity of the girl and sled together as v_final. The subscripts "girl" and "sled" help distinguish the two objects. The total momentum before the jump is the sum of the momentum of the girl and the momentum of the sled: Initial total momentum = (Mass of girl * v_girl_initial) + (Mass of sled * v_sled_initial) Substituting the given values: Initial total momentum = (50 kg * 1.5 m/s) + (8.5 kg * 0 m/s) [The sled is stationary initially] Initial total momentum = (75 kg·m/s) Now, after the girl jumps onto the sled, they move together as one system. The total momentum of the system after the jump should still be equal to the initial total momentum: Final total momentum = (Total Mass * v_final) The total mass of the system is the sum of the girl's mass and the sled's mass: Total Mass = Mass of girl + Mass of sled Total Mass = 50 kg + 8.5 kg Total Mass = 58.5 kg So, we have: Final total momentum = (58.5 kg * v_final) According to the conservation of momentum: Initial total momentum = Final total momentum (75 kg·m/s) = (58.5 kg * v_final) Now, solve for v_final: v_final = (75 kg·m/s) / (58.5 kg) v_final ≈ 1.28 m/s So, the sled with the girl moves at a velocity of approximately 1.28 m/s after she jumps onto it.