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 linear momentum. According to this principle, the total momentum before an event is equal to the total momentum after the event, assuming no external forces are acting on the system.
The formula for linear momentum is given by:
momentum = mass × velocity
Before the boy falls off the skateboard, the total momentum of the system is the sum of the individual momenta of the boy and the skateboard.
Total momentum before = (mass of boy × velocity of boy ) + (mass of skateboard × velocity of skateboard)
Initially, the boy and the skateboard are moving together with a velocity of 10.4 m/s. So their initial velocity is the same.
Total momentum before = ((mass of boy + mass of skateboard) × initial velocity)
After the boy falls off, the skateboard continues moving with a different velocity. Let's call this final velocity of the skateboard as Vf.
Total momentum after = (mass of boy × velocity of boy's center of mass) + (mass of skateboard × velocity of the skateboard)
According to the principle of conservation of linear momentum:
Total momentum before = Total momentum after
((mass of boy + mass of skateboard) × initial velocity) = (mass of boy × velocity of boy's center of mass) + (mass of skateboard × Vf)
Substituting the given values, we have:
((54.3 kg + 2.00 kg) × 10.4 m/s) = (54.3 kg × 10.9 m/s) + (2.00 kg × Vf)
Simplifying the equation, we can isolate Vf:
(56.3 kg × 10.4 m/s) - (54.3 kg × 10.9 m/s) = 2.00 kg × Vf
586.32 kg·m/s - 592.27 kg·m/s = 2.00 kg × Vf
-5.95 kg·m/s = 2.00 kg × Vf
Vf = -5.95 kg·m/s ÷ 2.00 kg
Vf = -2.975 m/s
The magnitude of the final velocity of the skateboard is 2.975 m/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 should equal the total momentum after he falls off.
The momentum of an object can be calculated by multiplying its mass by its velocity.
Initially, the total momentum is given by:
Total momentum before = (mass of the boy * velocity of the boy) + (mass of the skateboard * velocity of the skateboard)
= (54.3 kg * 10.4 m/s) + (2.00 kg * 10.4 m/s)
Let's calculate the initial total momentum:
Total momentum before = (565.32 kg*m/s) + (20.8 kg*m/s)
= 586.12 kg*m/s
After the boy falls off, his momentum will be carried by his center of mass, while the skateboard will have its own momentum.
So, the total momentum after the boy falls off is:
Total momentum after = (mass of the boy * velocity of the boy's center of mass) + (mass of the skateboard * velocity of the skateboard)
We are given the velocity of the boy's center of mass, but we need to find the velocity of the skateboard.
Using the conservation of momentum, we have:
Total momentum before = Total momentum after
586.12 kg*m/s = (mass of the boy * 10.9 m/s) + (mass of the skateboard * final velocity of the skateboard)
We can rearrange the equation to solve for the final velocity of the skateboard:
final velocity of the skateboard = (586.12 kg*m/s - (mass of the boy * 10.9 m/s)) / mass of the skateboard
Let's substitute the given values to find the final velocity of the skateboard:
final velocity of the skateboard = (586.12 kg*m/s - (54.3 kg * 10.9 m/s)) / 2.00 kg
final velocity of the skateboard = (586.12 kg*m/s - 592.47 kg*m/s) / 2.00 kg
final velocity of the skateboard = -6.35 kg*m/s / 2.00 kg
final velocity of the skateboard = -3.175 m/s
The negative sign indicates that the direction of the skateboard's final velocity is opposite to the original direction (since it is moving in the opposite direction to the boy's center of mass).
Therefore, the magnitude of the final velocity of the skateboard is 3.175 m/s.