Final answer:
In an elastic collision of two satellites with a relative speed of 0.250 m/s, the final relative velocity would be -0.250 m/s. The negative sign indicates the velocity is in the opposite direction post-collision, but the magnitude remains unchanged due to the laws of conservation of momentum and kinetic energy.
Step-by-step explanation:
The question pertains to an elastic collision between two manned satellites in space. The concept of conservation of momentum and conservation of kinetic energy are key to solving this problem, as both are conserved in an elastic collision. Given that the initial relative speed is 0.250 m/s and the masses of the satellites are 4.00 × 10³ kg and 7.50 × 10³ kg respectively, the final relative velocity can be found using the equations for an elastic collision:
m1•v1_initial + m2•v2_initial = m1•v1_final + m2•v2_final
0.5•m1•v1_initial² + 0.5•m2•v2_initial² = 0.5•m1•v1_final² + 0.5•m2•v2_final²
However, as the question is seeking only the final relative velocity and not individual velocities post-collision, we can simply state that for an elastic collision, the relative velocity between two objects after collision is the same in magnitude and opposite in direction to that before the collision.
Therefore, the final relative velocity after the satellites collide elastically would be -0.250 m/s; it has the same magnitude but is in the opposite direction.