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
23.975m/s
To find the final velocity of the skateboard, we can use the law of conservation of momentum. According to this law, the total momentum before the boy falls off the skateboard should be equal to the total momentum after he falls off.
The momentum of an object is defined as the product of its mass and velocity:
momentum = mass × velocity
Before the boy falls off, the total momentum is equal to the sum of the momentum of the boy and the momentum of the skateboard. Let's call this total momentum "P_total_initial".
P_total_initial = momentum of the boy + momentum of the skateboard
The momentum of the boy can be calculated as:
momentum of the boy = mass of the boy × velocity of the boy
Similarly, the momentum of the skateboard can be calculated as:
momentum of the skateboard = mass of the skateboard × velocity of the skateboard
Substituting these values into the equation for the total momentum:
P_total_initial = (mass of the boy × velocity of the boy) + (mass of the skateboard × velocity of the skateboard)
We are given the following values:
mass of the boy = 54.3 kg
velocity of the boy = 10.9 m/s
mass of the skateboard = 2.00 kg
velocity of the skateboard = unknown (let's call it "v_skateboard_final")
So the equation becomes:
P_total_initial = (54.3 kg × 10.9 m/s) + (2.00 kg × v_skateboard_final)
Now we know the initial total momentum of the system. After the boy falls off, the final total momentum of the system should be the same. The momentum of the boy after he falls off can be calculated as:
momentum of the boy = mass of the boy × velocity of the boy (after falling off)
Since the boy's center of mass moves forward at a speed of 10.9 m/s after falling off, we can substitute this value:
momentum of the boy = 54.3 kg × 10.9 m/s
Now we can write the equation for the final total momentum:
P_total_final = (54.3 kg × 10.9 m/s) + (2.00 kg × v_skateboard_final)
According to the law of conservation of momentum, P_total_initial should be equal to P_total_final:
P_total_initial = P_total_final
Substituting the values:
(54.3 kg × 10.9 m/s) + (2.00 kg × v_skateboard_final) = (54.3 kg × 10.9 m/s) + (2.00 kg × v_skateboard_final)
Simplifying the equation:
2.00 kg × v_skateboard_final = 0
This equation tells us that the final momentum of the skateboard is zero, which means the skateboard comes to rest after the boy falls off. Therefore, the final velocity of the skateboard is zero.