A 57.0-kg boy and his 43.0-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.
(b) How much potential energy in the girls body is converted into mechanical energy of the boy-girl system?
(c) Is the momentum of the boy-girl system conserved in the pushing-apart process? If so, explain how that is possible considering (d) there are large forces acting and
(e) there is no motion beforehand and plenty of motion afterward.

a) The motion of the girl opposes that of the boy

b) Mechanical Energy = 596.56 J

c) The momentum of the boy-girl system is conserved

Check explanations for d) and e)

Explanation:

Mass of the boy, M₁ = 57.0 kg

Mass of the sister, M₂ = 43.0 kg

Velocity of the boy, v₁ = 3.00 m/s

Velocity of the girl, v₂ = ?

For momentum to be conserved:

M₁v₁ = -M₂v₂

v₂ = M₁v₁/M₂

v₂ = (-57*3)/43

v₂ = -3.977 m/s

The motion of the girl is opposite in direction to the motion of the boy. Both of them have equal both opposite momentum

Momentum of the boy = M₁v₁ = 57 * 3

Momentum of the boy = 171 kgm/s²

Momentum of the girl = -M₂v₂ = - (43 * 3.977)

Momentum of the girl = -171 kgm/s²

b) The potential energy in the girl's body converted to mechanical energy of the boy-girl system is the summation of the individual kinetic energy of the boy and the girl

Mechanical Energy = 0.5(M₁v₁² + M₂v₂²)

Mechanical Energy = 0.5((57 * 3²) +(43 * 3.977²)

Mechanical Energy = 0.5(513 + 680.11)

Mechanical Energy = 596.56 J

c) The momentum is conserved because the velocities are opposite and the summation of the momenta is zero which obeys the principle of conservation of momentum.

d) Since both exert equal and opposite forces, one annuls the effect of the other. Newton's third law

e) Since the velocities are opposite and equal, momentum is conserved regardless of the initial speed.

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