Final answer:
The paintball will land approximately 11.3 meters from the base of the table. To determine how far from the base of the table the paintball will land, we can use the principle of conservation of momentum. We can set up an equation using the initial momentum of the paintball and the final momentum of the statue and paintball system:
Step-by-step explanation:
Momentum of paintball = Momentum of statue + Momentum of paintball and statue system after collision
Since the paintball is fired horizontally and collides with the statue, we can equate the initial momentum of the paintball to its final horizontal momentum:
(mass of paintball) * (initial velocity of paintball) = (mass of paintball) * (final velocity of paintball and statue system)
Using the given values, we can solve for the final velocity of the paintball and statue system:
(0.115 kg) * (540 m/s) = (0.115 kg + 6.8 kg) * (final velocity)
Simplifying the equation:
62.1 = 6.915 * (final velocity)
Dividing both sides by 6.915:
final velocity = 62.1 / 6.915
= 8.98 m/s
Now we can find the time it takes for the statue and paintball system to fall from the table height using the equation:
Height = (1/2) * (acceleration due to gravity) * (time squared)
Substituting the given values:
7.8 m = (0.5) * (9.8 m/s^2) * (time squared)
Simplifying the equation:
7.8 = 4.9 * (time squared)
Dividing both sides by 4.9:
time squared = 1.5918
Taking the square root of both sides:
time = 1.26 seconds
Using the final velocity of the paintball and statue system and the time, we can find the horizontal distance it travels:
Distance = (final velocity) * (time)
Substituting the given values:
Distance = (8.98 m/s) * (1.26 seconds)
Calculating the distance:
Distance = 11.3148 meters
Therefore, the paintball will land approximately 11.3 meters from the base of the table.