Two manned satellites approaching one another at a relative speed of 0.200 m/s intend to dock. The first has a mass of 2.00 multiplied by 103 kg, and the second a mass of 7.50 multiplied by 103 kg. If the two satellites collide elastically rather than dock, what is their final relative velocity? Adopt the reference frame in which the second satellite is initially at rest and assume that the positive direction is directed from the second satellite towards the first satellite.
To find the final relative velocity of the satellites after an elastic collision, we can use the principle of conservation of momentum.
The equation for conservation of momentum in a one-dimensional collision is:
m1 * v1i + m2 * v2i = m1 * v1f + m2 * v2f
Where:
- m1 and m2 are the masses of the satellites
- v1i and v2i are the initial velocities of the satellites
- v1f and v2f are the final velocities of the satellites
We are given that the second satellite is initially at rest, so v2i = 0.
Now we substitute the known values into the equation:
(2.00 × 10^3 kg) * v1i + (7.50 × 10^3 kg) * (0) = (2.00 × 10^3 kg) * v1f + (7.50 × 10^3 kg) * v2f
Simplifying the equation further:
2.00 × 10^3 kg * v1i = 2.00 × 10^3 kg * v1f + 7.50 × 10^3 kg * v2f
Since the two satellites are colliding elastically, we know that the total momentum before the collision should be equal to the total momentum after the collision. Therefore:
m1 * v1i + m2 * v2i = m1 * v1f + m2 * v2f
Now, we can solve for the final relative velocity, v1f - v2f:
v1f - v2f = (m1 * v1i + m2 * v2i - m2 * v2f) / m1
Substituting the known values:
v1f - v2f = (2.00 × 10^3 kg * v1i + 7.50 × 10^3 kg * 0 - 7.50 × 10^3 kg * v2f) / (2.00 × 10^3 kg)
Simplifying further:
v1f - v2f = (2.00 × 10^3 kg * v1i - 7.50 × 10^3 kg * v2f) / (2.00 × 10^3 kg)
Now, you can calculate the final relative velocity by plugging in the known values for the masses and initial velocity of the first satellite, which has a relative speed of 0.200 m/s.