Final answer:
Using the conservation of momentum and the work-energy principle, we find that Chance, the dog, is moving at a speed of 2.05 m/s when he meets his sister on the ice-covered sidewalk.
Step-by-step explanation:
To calculate how fast Chance is moving when he meets his sister, we can apply the principle of conservation of momentum. Since there are no external forces acting on the system of the sister and her dog (assuming a frictionless surface and ignoring air resistance), the momentum before the sister starts pulling Chance must equal the momentum when they meet.
Initially, both the sister and Chance are at rest, so the total initial momentum (Pi) is zero. When the sister exerts a force to pull Chance towards her, she will move backward while Chance moves toward her. The final momentum (Pf) just before they meet will still be zero because the actions are equal and opposite due to Newton's third law.
Let's denote the final velocity of Chance as VChance and the final velocity of the sister as VSister. Since momentum is conserved:
M1 * VSister + M2 * VChance = 0
VSister = - (M2 / M1) * VChance
The total work done by the sister on Chance due to the force exerted will be equal to the change in kinetic energy of Chance:
Work = Force * distance = F * d
Change in kinetic energy (ΔKE) = 0.5 * M2 * VChance2 - 0
F * d = 0.5 * M2 * VChance2
VChance = √((2 * F * d) / M2)
VChance is the velocity of Chance when he meets his sister. Substituting the values, we get:
VChance = √((2 * 20.0 N * 4.0 m) / 38 kg)
VChance = √((160 N*m) / 38 kg)
VChance = √(4.2105 m2/s2)
VChance = 2.05 m/s
Thus, Chance is moving at a speed of 2.05 m/s when he meets his sister.