During July 1995 the comet Shemaker-Levy 9 smashed into Jupiter in a spectacular fashion. The comet actually consisted of 21 distinct pieces, the largest of which had a mass of about 4x1012 kg and a speed of 6x104 m/s. Jupiter, the largest planet in the solar system , has a mass of 1.9x1027 kg and an orbital speed of 1.3x104 m/s. If this piece of the comet had hit Jupiter head on , what would have been the change (magnitude only) in Jupiter's orbital speed(not its final speed)?
To calculate the change in Jupiter's orbital speed if the comet piece had hit it head-on, we can use the principle of conservation of momentum.
The principle of conservation of momentum states that the total momentum of an isolated system remains constant unless acted upon by external forces. In this case, we can consider the comet piece and Jupiter as an isolated system.
The momentum of an object is calculated by multiplying its mass and velocity (p = m * v).
Before the collision, the total momentum of the system is given by the sum of the momenta of the comet piece and Jupiter:
Total initial momentum = (mass of comet piece * velocity of comet piece) + (mass of Jupiter * velocity of Jupiter)
Total initial momentum = (4x10^12 kg * 6x10^4 m/s) + (1.9x10^27 kg * 1.3x10^4 m/s)
Since the comet piece is hitting Jupiter head-on, we can assume their relative velocities add up. Therefore, the velocity of the comet piece relative to Jupiter is:
Relative velocity = velocity of comet piece - velocity of Jupiter
Relative velocity = 6x10^4 m/s - 1.3x10^4 m/s
Now, let's calculate the change in momentum of the system due to the collision:
Change in momentum = (mass of comet piece * relative velocity)
Change in momentum = (4x10^12 kg) * (6x10^4 m/s - 1.3x10^4 m/s)
Finally, to find the change in Jupiter's orbital speed, we need to divide the change in momentum by the mass of Jupiter:
Change in Jupiter's orbital speed = Change in momentum / mass of Jupiter
Change in Jupiter's orbital speed = (4x10^12 kg * (6x10^4 m/s - 1.3x10^4 m/s)) / 1.9x10^27 kg
Simplifying the equation will give you the answer, which represents the change in magnitude of Jupiter's orbital speed if the comet piece had hit it head-on during the collision.
Please note that the result we obtain is the change in Jupiter's orbital speed and not its final speed after the collision.