Final answer:
The velocity of the 31.0-g object (Object 2) after the perfectly elastic collision with the 13.5-g object (Object 1) moving to the right with a speed of 20.4 cm/s is calculated to be approximately 12.31 cm/s.
Step-by-step explanation:
To calculate the velocity of the 31.0-g object (Object 2) after a perfectly elastic collision with a 13.5-g object (Object 1), we can use the principles of conservation of momentum and conservation of kinetic energy.
The initial momentum of the system is given by the momentum of Object 1 before the collision since Object 2 is at rest.
Momentum before collision = Momentum after collision
(13.5 g × 20.4 cm/s) + (31.0 g × 0 cm/s) = (13.5 g × final velocity of Object 1) + (31.0 g × final velocity of Object 2)
As the collision is perfectly elastic, kinetic energy is also conserved:
KE before collision = KE after collision
(1/2)×(13.5 g)×(20.4 cm/s)² = (1/2)×(13.5 g)×(final velocity of Object 1)² + (1/2)×(31.0 g)×(final velocity of Object 2)²
By solving these equations simultaneously, we can find the final velocities of both objects. For object 2:
Final velocity of Object 1 (v1') = (m1 - m2)/(m1 + m2) × initial velocity of Object 1
Final velocity of Object 2 (v2') = (2×m1)/(m1 + m2) × initial velocity of Object 1
Final velocity of Object 2 (v2') = (2× 13.5 g)/(13.5 g + 31.0 g) × 20.4 cm/s
Final velocity of Object 2 (v2') = 27 g/44.5 g × 20.4 cm/s
Final velocity of Object 2 (v2') = 12.31 cm/s