← Back to Concepts

Collision Examples

Collisions can be complex when they involve both translational and rotational motion. Off-center impacts and spin-inducing collisions demonstrate how angular momentum and linear momentum interact during collisions.

Off-center Impact on a Rod

When a particle strikes a rod off-center, the collision transfers both linear momentum and angular momentum to the rod. This creates a combination of translational and rotational motion.

Key Equations:
Linear momentum conservation: \(m_1 v_{1i} = m_1 v_{1f} + m_2 v_{2f}\)
Angular momentum conservation: \(m_1 v_{1i} d = m_1 v_{1f} d + I_2 \omega_2\)
Rod moment of inertia: \(I = \frac{1}{12}ML^2\) (about center)
Kinetic energy: \(K = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2\)

Off-center Impact Simulation

Particle Mass (kg):

Current: 0.5 kg

Particle Velocity (m/s):

Current: 10 m/s

Impact Position:

Current: 0.3 (fraction of rod length)

Collision Analysis:
Initial Momentum: 5.0 kg⋅m/s
Final Linear Velocity: 0.0 m/s
Final Angular Velocity: 0.0 rad/s

Energy Distribution:

Translational Kinetic: 0.0 J

Rotational Kinetic: 0.0 J

Example: Off-center Impact Calculation

A 0.5 kg particle moving at 10 m/s strikes a 2 kg rod (length 1 m) at 0.3 of its length from the center. What are the final velocities?

Solution:

Momentum conservation: \(0.5 \times 10 = 0.5 v_{1f} + 2 v_{2f}\)

Angular momentum conservation: \(0.5 \times 10 \times 0.3 = 0.5 v_{1f} \times 0.3 + \frac{1}{12} \times 2 \times 1^2 \times \omega\)

Solving: \(v_{2f} = 2.4\) m/s, \(\omega = 18\) rad/s

Billiard Ball Spin

When a billiard ball is struck off-center, it acquires both translational velocity and angular velocity (spin). The spin affects the ball's trajectory and collision behavior.

Key Equations:
Linear momentum: \(p = mv\)
Angular momentum: \(L = I\omega = \frac{2}{5}mr^2\omega\) (solid sphere)
Rolling condition: \(v = \omega r\) (when rolling without slipping)
Kinetic energy: \(K = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2\)

Billiard Ball Spin Simulation

Cue Velocity (m/s):

Current: 8.0 m/s

Spin Type:

Spin Intensity:

Current: 0.5 (0=no spin, 1=max spin)

Table Friction:

Current: 0.3

Ball Motion Analysis:
Linear Velocity: 0.0 m/s
Angular Velocity: 0.0 rad/s
Spin Type: None

Energy Distribution:

Translational Kinetic: 0.0 J

Rotational Kinetic: 0.0 J

Types of Billiard Ball Spin

Spin Type	Description	Effect on Motion	Application
Top Spin	Forward rotation	Ball rolls faster, longer distance	Power shots, long distance
Back Spin	Backward rotation	Ball slows down, may reverse direction	Control shots, stopping
Side Spin	Lateral rotation	Ball curves sideways	Angle shots, banking
No Spin	Pure translation	Ball slides, then rolls	Straight shots

Example: Billiard Ball Spin Calculation

A billiard ball (mass 0.17 kg, radius 0.028 m) is struck with a cue at 8 m/s with 30% spin factor. What are the initial translational and angular velocities?

Solution:

Translational velocity: \(v = 8\) m/s

Angular velocity: \(\omega = \frac{v \times \text{spin factor}}{r} = \frac{8 \times 0.3}{0.028} = 85.7\) rad/s

Translational energy: \(K_t = \frac{1}{2} \times 0.17 \times 8^2 = 5.44\) J

Rotational energy: \(K_r = \frac{1}{2} \times \frac{2}{5} \times 0.17 \times 0.028^2 \times 85.7^2 = 0.98\) J

Real-World Applications

Sports: Understanding spin in tennis, golf, and billiards
Automotive: Tire grip and vehicle stability
Engineering: Gear design and mechanical systems
Physics Research: Particle collisions and angular momentum conservation

Summary / Takeaways

Off-center collisions transfer both linear and angular momentum
Spin affects the trajectory and behavior of rolling objects
Energy is distributed between translational and rotational motion
Friction plays a crucial role in converting sliding to rolling motion
Understanding spin is essential for many sports and engineering applications