Final answer:
To determine the mechanical energy dissipated in a ballistic pendulum collision, one must calculate the final velocity of the system after the bullet embeds into the pendulum bob using conservation of momentum. The difference in kinetic energy before and after the collision gives the dissipated energy. Additionally, with given specific heat capacity and molecular mass, one can calculate the temperature increase of the system.
Step-by-step explanation:
Projectile Motion and Energy Dissipation in a Ballistic Pendulum
A bullet of mass 10 g is traveling horizontally at 200 m/s and embeds in a pendulum bob of mass 2.0 kg, thus forming a classic example of a ballistic pendulum problem in physics. We are tasked with finding the amount of mechanical energy dissipated during the collision and calculating the temperature increase of the system caused by the collision.
First, we need to determine the final velocity of the system (bullet plus pendulum bob) immediately after the collision, as this will directly tie into the energy dissipated. Conservation of momentum can be applied since no external forces are acting horizontally during the collision:
m_bullet * v_initial_bullet = (m_bullet + m_bob) * v_final_system,
where m_bullet is the mass of the bullet, v_initial_bullet is its initial velocity, and v_final_system is the final velocity of the combined system. This final velocity is needed to figure out the kinetic energy before and after the collision.
The kinetic energy lost (mechanical energy dissipated) is the difference between the initial kinetic energy of the bullet and the final kinetic energy of the combined system:
Energy_dissipated = (1/2) * m_bullet * v_initial_bullet^2 - (1/2) * (m_bullet + m_bob) * v_final_system^2.
Since the molecular mass of the system and a specific heat capacity (C₁) is given, we can calculate the temperature increase using the formula:
Temperature_increase = Energy_dissipated / (m_total * C₁),
where m_total is the total mass of bullet and bob, and C₁ is the specific heat capacity. Remember to convert the mass to kilograms before performing the calculation.