A boy is standing on a skateboard holding two 5.0 kg bowling balls. Initially they are moving to the left at a speed of 3.0 m/s. At one point the boy throws one ball to the right at 4.0 m/s causing his speed to increase. Before he can slow down, he throws the second ball to the right at 4.0 m/s. If the boy and the board have a mass of 45 kg, what is the velocity immediatly after throwing the second ball?
To find the final velocity immediately after throwing the second ball, we need to use the principle of conservation of momentum. According to this principle, the total momentum before and after the ball is thrown must be the same.
The momentum of an object is defined as the product of its mass and velocity. So, let's calculate the total momentum before and after throwing the second ball:
Initial momentum before throwing any ball:
Momentum of boy and board = (mass of boy and board) × (initial velocity of boy and board)
= 45 kg × (-3.0 m/s)
= -135 kg·m/s (negative because the initial velocity is to the left)
Now, after throwing the first ball, let's calculate the momentum:
Momentum of boy and board = (mass of boy and board) × (final velocity of boy and board)
= 45 kg × (final velocity of boy and board)
Since the first ball is thrown to the right, its momentum is in the positive direction. So, the total momentum after throwing the first ball is:
Total momentum = Momentum of boy and board + Momentum of first ball
= 45 kg × (final velocity of boy and board) + (5.0 kg × 4.0 m/s)
Now, after throwing the second ball, let's calculate the momentum:
Momentum of boy and board = (mass of boy and board) × (final velocity of boy and board)
= 45 kg × (final velocity of boy and board)
Since the second ball is also thrown to the right, its momentum is in the positive direction. So, the total momentum after throwing the second ball is:
Total momentum = Momentum of boy and board + Momentum of first ball + Momentum of second ball
= 45 kg × (final velocity of boy and board) + (5.0 kg × 4.0 m/s) + (5.0 kg × 4.0 m/s)
According to the principle of conservation of momentum, the total momentum before and after throwing the ball must be the same. Therefore, we can equate the two equations:
-135 kg·m/s = 45 kg × (final velocity of boy and board) + (5.0 kg × 4.0 m/s) + (5.0 kg × 4.0 m/s)
Now, let's solve this equation to find the final velocity of the boy and the board.