Final answer:
Without the initial speed of the ball before the collision, it's not possible to calculate the final speed of the ball and cup after the collision using the provided information. The conservation of momentum cannot be applied to find the final velocity without this critical piece of data.
Step-by-step explanation:
The question involves the concept of conservation of momentum. When the ball and cup have moved 5.0 cm after collision, the speed can be found using the principle that momentum before the collision equals momentum after the collision, as long as there are no external forces. Since the masses of the ball and cup are given in grams, they should first be converted to kilograms by dividing by 1000 (82.0 grams = 0.082 kg, 278.8 grams = 0.2788 kg). Let's denote m1 as the mass of the ball and m2 as the mass of the cup, v1 as the initial velocity of the ball, and v' as the final velocity of both after the collision. According to the conservation of momentum m1 × v1 = (m1 + m2) × v' To find the final velocity v', we rearrange the equation: v' = (m1 × v1) / (m1 + m2) Since the question does not provide the initial speed of the ball, it isn't possible to calculate the final speed of the combined ball and cup without making assumptions or having further information. However, if we assume the ball was at rest initially like the cup, then the final velocity would be 0 m/s because both would remain at rest, indicating no movement occurred. Hence, with the given information, we cannot determine the speed just after the collision.