Question

How much potential energy in the girl's body is converted into mechanical energy of the boy–girl system

Answer 1

The complete question is:

A 65 kg boy and his 40 kg sister, both wearing roller blades, face each other at rest. The girl pushes the boy hard, sending him backward with a velocity 2.90 m/s toward the west. Ignore friction."

How much potential energy in the girl's body is converted into mechanical energy of the boy–girl system?

Answer:

The ammount of potential energy converted to mechanical energy is 717 Joules

Explanation:

Knowing the Law of Momentum Conservation we can state that the momentum change of the boy is equal to and opposite of the momentum change of the girl.

Written in a formula:

$m_{boy} *v_{boy} =-m_{girl} *v_{girl}$

where

$m_{boy}\: \textup{is the boy's mass}\\v_{boy} \: \textup{is the boy's velocity}\\m_{girl} \: \textup{is the girl's mass}\\v_{girl} \: \textup{is the girl's velocity}$

Solving for $v_{girl}$:

$v_{girl}=\frac{m_{boy} * v_{boy}}{-m_{girl}}\\v_{girl}=\frac{65kg * 2.9\frac{m}{s} }{-40kg}\\v_{girl}=-4.71\frac{m}{s}$

Finally, using the energy conservation formula:

$\Delta K_{energy} = \frac{1}{2}*m_{boy}* v_{boy}^{2} + \frac{1}{2}*m_{girl}* v_{girl}^{2}\\\Delta K_{energy} = \frac{1}{2}*65kg* (2.9\frac{m}{s})^{2} + \frac{1}{2}*40kg* (-4.71\frac{m}{s})^{2}\\\Delta K_{energy} = 273.325 \frac{kg*m^{2}}{s^{2}}+ 443.682\frac{kg*m^{2}}{s^{2}}\\\Delta K_{energy} = 717\frac{kg*m^{2}}{s^{2}} = 717 J$