10.6 Collisions of Extended Bodies in Two Dimensions

Collisions between extended bodies in two dimensions involve both linear and angular momentum. These complex interactions require us to consider not just the motion of objects as a whole, but also their rotation and internal structure.

Understanding these collisions helps explain real-world phenomena, from sports to car crashes. We'll explore how energy changes during collisions and the role of special points like the center of percussion in determining collision outcomes.

Collisions of Extended Bodies in Two Dimensions

Collisions of extended bodies

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• Conservation of linear momentum applies to collisions in two dimensions, where the total linear momentum before and after the collision remains constant, represented by the equation $m_1 \vec{v}_1 + m_2 \vec{v}_2 = m_1 \vec{v'}_1 + m_2 \vec{v'}_2$
  • $m_1, m_2$ represent the masses of the colliding objects (billiard balls)
  • $\vec{v}_1, \vec{v}_2$ denote the initial velocities before the collision
  • $\vec{v'}_1, \vec{v'}_2$ represent the final velocities after the collision
• Conservation of angular momentum applies to collisions involving extended bodies, where the total angular momentum before and after the collision remains constant, given by the equation $I_1 \omega_1 + I_2 \omega_2 = I_1 \omega'_1 + I_2 \omega'_2$
  • $I_1, I_2$ represent the moments of inertia of the colliding objects (spinning tops)
  • $\omega_1, \omega_2$ denote the initial angular velocities before the collision
  • $\omega'_1, \omega'_2$ represent the final angular velocities after the collision
• Impulse $\vec{J}$ equals the change in linear momentum $\Delta \vec{p}$, calculated using the equation $\vec{J} = \Delta \vec{p} = m (\vec{v'} - \vec{v})$, where $m$ is the mass and $\vec{v}, \vec{v'}$ are the initial and final velocities
• Change in angular momentum $\Delta \vec{L}$ is determined by the cross product of the position vector $\vec{r}$ from the axis of rotation to the point of collision and the impulse $\vec{J}$, expressed as $\Delta \vec{L} = \vec{r} \times \vec{J}$
  • The magnitude of this change is related to the torque applied during the collision

Percussion point applications

• Percussion point, also known as the center of percussion, is the point on an extended body where an impulse can be applied without causing rotation
• Located at a distance $R$ from the center of mass, calculated using the formula $R = \frac{I}{mh}$, where $I$ is the moment of inertia about the axis of rotation, $m$ is the mass of the object, and $h$ is the distance between the axis of rotation and the center of mass
• Baseball bats have their percussion point located near the "sweet spot," which minimizes vibrations and maximizes energy transfer to the ball upon impact
• Tennis rackets have their percussion point located near the center of the strings, reducing twisting of the racket when hitting the ball
• Golf clubs have their percussion point located on the clubface, helping achieve a straight shot without unwanted clubhead rotation

Energy changes in rotating collisions

• Rotational kinetic energy $K_r$ is calculated using the formula $K_r = \frac{1}{2} I \omega^2$, where $I$ is the moment of inertia and $\omega$ is the angular velocity
• Change in rotational kinetic energy $\Delta K_r$ is determined by the equation $\Delta K_r = \frac{1}{2} I (\omega'^2 - \omega^2)$, where $\omega$ and $\omega'$ are the initial and final angular velocities
• Linear kinetic energy $K_l$ is calculated using the formula $K_l = \frac{1}{2} m v^2$, where $m$ is the mass and $v$ is the linear velocity
• Change in linear kinetic energy $\Delta K_l$ is determined by the equation $\Delta K_l = \frac{1}{2} m (v'^2 - v^2)$, where $v$ and $v'$ are the initial and final linear velocities
• Total change in kinetic energy $\Delta K$ is the sum of the changes in rotational and linear kinetic energy, expressed as $\Delta K = \Delta K_r + \Delta K_l$
• Elastic collisions conserve total kinetic energy, while inelastic collisions do not conserve total kinetic energy due to energy dissipation through deformation or heat (car crashes)
  • The coefficient of restitution characterizes the elasticity of a collision, ranging from 0 (perfectly inelastic) to 1 (perfectly elastic)

Additional considerations in extended body collisions

• Center of mass plays a crucial role in analyzing collisions of extended bodies, as it represents the point where the entire mass of the object can be considered concentrated
• Friction between colliding surfaces can affect the outcome of a collision by introducing additional forces and energy dissipation
• Angular momentum conservation becomes particularly important when dealing with rotating extended bodies, as it influences both linear and rotational motion after the collision