A 65kg boy and his 40kg sister, both wearing roller blades, face each other at rest. The girl pushes the boy hard, sending him backwards with a velocity of 2.9m/s towards the west. Ignore friction. What is the minimum amount of chemical energy is converted into mechanical energy in the girl’s muscles?
It is easy to calculate how much kinetic energy must be given to both brother and sister. The girl acquires an equal momentum in the opposite direction, with velocity 2.9*64/40 = 4.64 m/s.
Calculate and add the kinetic energies of both brother and sister.
Chemical (food) energy cannot be converted to mechanical energy with 100% efficiency. The efficiency varies from person to person, but is around 20%. They may want you to assume 100% efficiency for the minimum food energy required, but the real number is much higher.
To find the minimum amount of chemical energy converted into mechanical energy in the girl's muscles, we need to consider the conservation of momentum.
The initial momentum of the system is zero since both the boy and the girl are initially at rest. The final momentum after the girl pushes the boy can be calculated using the formula:
Final momentum = (mass of the boy x velocity of the boy) + (mass of the girl x velocity of the girl)
We are given:
Mass of the boy (m₁) = 65 kg
Mass of the girl (m₂) = 40 kg
Velocity of the boy (v₁) = -2.9 m/s (negative sign indicates the boy is moving towards the west)
Velocity of the girl (v₂) = 0 m/s (since she is at rest)
Using these values, we can calculate the final momentum:
Final momentum = (65 kg x -2.9 m/s) + (40 kg x 0 m/s)
Final momentum = -188.5 kg⋅m/s
Since momentum is conserved, the change in momentum is equal to the impulse applied by the girl on the boy. In this case, the impulse is the difference in momentum:
Change in momentum (impulse) = Final momentum - Initial momentum
Change in momentum (impulse) = -188.5 kg⋅m/s - 0 kg⋅m/s
Change in momentum (impulse) = -188.5 kg⋅m/s
The impulse is equal to the change in mechanical energy. Therefore, the minimum amount of chemical energy converted into mechanical energy in the girl's muscles is equal to the magnitude of this impulse (since it is negative):
Minimum amount of chemical energy converted into mechanical energy = |Change in momentum|
Minimum amount of chemical energy converted into mechanical energy = |-188.5 kg⋅m/s|
Hence, the minimum amount of chemical energy converted into mechanical energy in the girl's muscles is 188.5 kg⋅m/s, or simply 188.5 Joules.
To find the minimum amount of chemical energy converted into mechanical energy in the girl's muscles, we need to use the principle of conservation of momentum.
First, let's calculate the total initial momentum before the push. The momentum is given by the equation:
initial momentum = mass × velocity
For the boy:
Boy's initial momentum = 65 kg × 0 m/s (since he is at rest)
Boy's initial momentum = 0 kg·m/s
For the girl:
Girl's initial momentum = 40 kg × 0 m/s (since she is at rest)
Girl's initial momentum = 0 kg·m/s
Therefore, the total initial momentum of the system is 0 kg·m/s.
According to the conservation of momentum, the total momentum before the push should be equal to the total momentum after the push. Since the boy moves backward with a velocity of 2.9 m/s towards the west, the girl should move forward with the same velocity to maintain the total momentum of the system at 0 kg·m/s.
Now, let's calculate the change in momentum of the girl. Since the girl is moving from rest, her final momentum will be:
Girl's final momentum = girl's mass × girl's final velocity
= 40 kg × 2.9 m/s (since the velocity is towards the east)
Girl's final momentum = 116 kg·m/s
To calculate the change in momentum, we subtract the initial momentum from the final momentum:
Change in momentum = Girl's final momentum - Girl's initial momentum
Change in momentum = 116 kg·m/s - 0 kg·m/s
Change in momentum = 116 kg·m/s
The change in momentum is equal to the force exerted by the girl on the boy. Since the force is acting over a distance (the boy's displacement), work is done.
To calculate the minimum amount of chemical energy converted into mechanical energy in the girl's muscles, we can use the work-energy principle. The work done is equal to the change in kinetic energy:
Work done = change in kinetic energy
Work done = (1/2) × (mass × velocity^2)
Work done = (1/2) × 65 kg × (2.9 m/s)^2
Work done = 582.45 J
Therefore, the minimum amount of chemical energy converted into mechanical energy in the girl's muscles is 582.45 Joules.