A boy of mass 54.3 kg is initially on a skateboard of mass 2.00 kg, moving at a speed of 10.4 m/s. The boy falls off the skateboard, and his center of mass moves forward at a speed of 10.9 m/s. Find the final velocity of the skateboard.
magnitude
To find the final velocity of the skateboard, we can use the principle of conservation of momentum, which states that the total momentum before an event is equal to the total momentum after the event, assuming there are no external forces acting.
The momentum of an object is calculated by multiplying its mass by its velocity. Therefore, the initial momentum (before the boy falls off) is given by:
Initial momentum = (Mass of boy on skateboard) * (Initial velocity of the skateboard)
= (54.3 kg + 2.00 kg) * (10.4 m/s)
The final momentum (after the boy falls off) is given by:
Final momentum = (Mass of boy) * (Final velocity of the boy) + (Mass of skateboard) * (Final velocity of the skateboard)
= (54.3 kg) * (10.9 m/s) + (2.00 kg) * (Final velocity of the skateboard)
Since the total momentum is conserved, we can equate the initial and final momentum:
(54.3 kg + 2.00 kg) * (10.4 m/s) = (54.3 kg) * (10.9 m/s) + (2.00 kg) * (Final velocity of the skateboard)
Now, let's solve for the final velocity of the skateboard:
(56.3 kg) * (10.4 m/s) = (54.3 kg) * (10.9 m/s) + (2.00 kg) * (Final velocity of the skateboard)
585.52 kg·m/s = 592.87 kg·m/s + (2.00 kg) * (Final velocity of the skateboard)
Subtracting 592.87 kg·m/s from both sides:
585.52 kg·m/s - 592.87 kg·m/s = (2.00 kg) * (Final velocity of the skateboard)
-7.35 kg·m/s = (2.00 kg) * (Final velocity of the skateboard)
Dividing both sides by 2.00 kg:
-3.675 m/s = Final velocity of the skateboard
Therefore, the final velocity of the skateboard is -3.675 m/s. The negative sign indicates that the skateboard is moving in the opposite direction to the initial motion.