A boy of mass 49.4 kg is initially on a skateboard of mass 2.00 kg, moving at a speed of 10.3 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.
To find the final velocity of the skateboard, we can use the principle of conservation of momentum, which states that the total momentum of a system remains constant if no external forces act on it.
The momentum is calculated by multiplying the mass of an object by its velocity:
Momentum = Mass x Velocity
Initial momentum of the system = (mass of the boy + mass of the skateboard) x initial velocity of the system
Final momentum of the system = (mass of the boy x final velocity of the boy) + (mass of the skateboard x final velocity of the skateboard)
Since the total momentum is conserved, we can set the initial momentum equal to the final momentum:
(mass of the boy + mass of the skateboard) x initial velocity of the system = (mass of the boy x final velocity of the boy) + (mass of the skateboard x final velocity of the skateboard)
Plugging in the given values:
(49.4 kg + 2.00 kg) x 10.3 m/s = (49.4 kg x 10.9 m/s) + (2.00 kg x final velocity of the skateboard)
Simplifying the equation:
(51.4 kg) x 10.3 m/s = (538.36 kg⋅m/s) + (2.00 kg x final velocity of the skateboard)
520.84 kg⋅m/s = 538.36 kg⋅m/s + (2.00 kg x final velocity of the skateboard)
Rearranging the equation to solve for the final velocity of the skateboard:
(final velocity of the skateboard) = (520.84 kg⋅m/s - 538.36 kg⋅m/s) / 2.00 kg
(final velocity of the skateboard) = -8.26 kg⋅m/s / 2.00 kg
(final velocity of the skateboard) = -4.13 m/s
Therefore, the final velocity of the skateboard is -4.13 m/s.
To find the final velocity of the skateboard, we can use the principle of conservation of momentum. According to this principle, the total momentum before and after the boy falls off the skateboard should be equal.
The momentum of an object is given by the product of its mass and velocity. Therefore, the initial momentum of the system (boy + skateboard) can be calculated as follows:
Initial momentum = (mass of the boy + mass of the skateboard) * initial velocity
= (49.4 kg + 2.00 kg) * 10.3 m/s
= 514.1 kg·m/s
After the boy falls off the skateboard, the total momentum will be the sum of the momentum of the boy and the momentum of the skateboard. The boy's momentum can be calculated by multiplying his mass by his final velocity (since his center of mass is moving forward):
Momentum of the boy = mass of the boy * final velocity of the boy
= 49.4 kg * 10.9 m/s
= 537.46 kg·m/s
Now, we can calculate the final momentum of the skateboard by subtracting the momentum of the boy from the initial momentum:
Final momentum of the skateboard = Initial momentum - Momentum of the boy
= 514.1 kg·m/s - 537.46 kg·m/s
= -23.36 kg·m/s
Since momentum is a vector quantity, the negative sign indicates that the skateboard is moving in the opposite direction compared to the initial velocity of the system.
Finally, we can find the final velocity of the skateboard by dividing the final momentum by its mass:
Final velocity of the skateboard = Final momentum of the skateboard / mass of the skateboard
= (-23.36 kg·m/s) / 2.00 kg
= -11.68 m/s
Therefore, the final velocity of the skateboard is -11.68 m/s. The negative sign indicates that the skateboard is moving in the opposite direction compared to its initial velocity.