A 53.0 kg boy and his 41.0 kg sister, both wearing roller blades face each other at rest. The girl pushes the boy hard, sending him backward with a velocity 3.00 m/s toward the west. Ignore friction.
(a) Describe the subsequent motion of the girl, and then check it against the posted solution after the assignment is due. Although this will not count directly on your grade, it will be good preparation for the test. How much chemical energy is converted into mechanical energy in the girl's muscles?
(a) Use conservation of momentum.
The mechanical energy converted by the girl equals the increase in the combined kinetic energy of both
When the girl pushes the boy, according to Newton's third law of motion, an equal and opposite force acts on the girl. This means that the girl will experience a push in the opposite direction, i.e., towards the east.
Since there is no friction, both the boy and the girl will continue to move with a constant velocity after the push. The boy will move with a velocity of 3.00 m/s towards the west, while the girl will move with the same magnitude of velocity, 3.00 m/s towards the east.
To calculate the amount of chemical energy converted into mechanical energy in the girl's muscles, we need to use the conservation of mechanical energy, which states that the initial mechanical energy before the push is equal to the final mechanical energy after the push.
The initial mechanical energy is given by the sum of the kinetic energies of the boy and the girl:
Initial mechanical energy = (1/2) * m_boy * v_boy^2 + (1/2) * m_girl * v_girl^2
The final mechanical energy is zero, as both the boy and the girl will eventually come to a stop.
Therefore, the amount of chemical energy converted into mechanical energy in the girl's muscles can be calculated by subtracting the initial mechanical energy from zero:
Chemical energy converted into mechanical energy = 0 - Initial mechanical energy
I apologize, but I am unable to provide numerical calculations.
To describe the subsequent motion of the girl, we can apply the law of conservation of momentum. According to this law, the total momentum before an event is equal to the total momentum after the event.
Initial total momentum = Final total momentum
Before the girl pushes the boy, both of them are at rest, so their initial momenta are zero.
Initial total momentum = 0
After the girl pushes the boy, the boy moves backward with a velocity of 3.00 m/s toward the west. To maintain the conservation of momentum, the girl must move in the opposite direction with a velocity that depends on their masses.
From the problem statement, the mass of the boy is 53.0 kg and the mass of the girl is 41.0 kg.
The momentum of an object is given by the product of its mass and velocity. So, we can calculate the initial momentum of the boy, considering that his velocity is backward (toward the west):
Initial momentum of the boy = mass of the boy * velocity of the boy
= 53.0 kg * (-3.00 m/s)
= -159.0 kg·m/s (negative sign indicates its direction)
Since the initial total momentum is zero, the initial momentum of the girl must be positive and equal to 159.0 kg·m/s:
Initial momentum of the girl = 159.0 kg·m/s
Now, after the girl pushes the boy, she will move backward to conserve momentum. The velocity of the girl can be calculated using the mass of the girl and the initial total momentum:
Final velocity of the girl = Initial total momentum / mass of the girl
= 159.0 kg·m/s / 41.0 kg
≈ 3.88 m/s
Therefore, the subsequent motion of the girl is that she moves backward with a velocity of 3.88 m/s.
Regarding the conversion of chemical energy to mechanical energy in the girl's muscles, the problem does not provide information about the efficiency of this conversion or the amount of chemical energy initially stored in the girl's muscles. Without this information, it is not possible to determine the exact amount of energy converted.