Elastic Collision Formula Calculator

\[ v_{1f} = \frac{m_1 - m_2}{m_1 + m_2} v_{1i} + \frac{2 m_2}{m_1 + m_2} v_{2i} \] \[ v_{2f} = \frac{2 m_1}{m_1 + m_2} v_{1i} + \frac{m_2 - m_1}{m_1 + m_2} v_{2i} \]

Mass of Object 1 (\(m_1\)):

Initial Velocity of Object 1 (\(v_{1i}\)):

Mass of Object 2 (\(m_2\)):

Initial Velocity of Object 2 (\(v_{2i}\)):

Final Velocity of Object 1 (\(v_{1f}\)):

Final Velocity of Object 2 (\(v_{2f}\)):

m/s

1. What is the Elastic Collision Formula Calculator?

Definition: This calculator computes the final velocities (\(v_{1f}\), \(v_{2f}\)) of two objects after a one-dimensional elastic collision, given their masses (\(m_1\), \(m_2\)) and initial velocities (\(v_{1i}\), \(v_{2i}\)).

Purpose: It is used in physics to analyze collisions where both momentum and kinetic energy are conserved, applicable in scenarios like billiard balls, particle interactions, and mechanical systems.

2. How Does the Calculator Work?

The calculator uses the elastic collision formulas derived from conservation of momentum and kinetic energy:

Formulas: \[ v_{1f} = \frac{m_1 - m_2}{m_1 + m_2} v_{1i} + \frac{2 m_2}{m_1 + m_2} v_{2i} \] \[ v_{2f} = \frac{2 m_1}{m_1 + m_2} v_{1i} + \frac{m_2 - m_1}{m_1 + m_2} v_{2i} \] where:

\(v_{1f}\), \(v_{2f}\): Final velocities of objects 1 and 2 (m/s, km/s, ft/s, yd/s)
\(m_1\), \(m_2\): Masses of objects 1 and 2 (kg, g, lb, oz)
\(v_{1i}\), \(v_{2i}\): Initial velocities of objects 1 and 2 (m/s, km/s, ft/s, yd/s)

Unit Conversions:

Mass:
- 1 kg = 1 kg
- 1 g = 0.001 kg
- 1 lb = 0.45359237 kg
- 1 oz = 0.028349523125 kg
Velocity (Initial and Final):
- 1 m/s = 1 m/s
- 1 km/s = 1000 m/s
- 1 ft/s = 0.3048 m/s
- 1 yd/s = 0.9144 m/s

Steps:

Enter the masses (\(m_1\), \(m_2\)) and initial velocities (\(v_{1i}\), \(v_{2i}\)) with their units (default: \(m_1 = 1 \, \text{kg}\), \(v_{1i} = 5 \, \text{m/s}\), \(m_2 = 2 \, \text{kg}\), \(v_{2i} = 0 \, \text{m/s}\)).
Convert inputs to SI units (kg, m/s).
Validate that masses are greater than 0 and the sum of masses is non-zero.
Calculate the final velocities using the elastic collision formulas.
Convert the final velocities to the selected unit (m/s, km/s, ft/s, or yd/s).
Display the results, rounded to 4 decimal places.

3. Importance of Elastic Collision Calculation

Calculating the outcomes of elastic collisions is crucial for:

Physics: Understanding momentum and energy conservation in idealized systems like atomic collisions or billiard games.
Engineering: Analyzing impacts in mechanical systems, such as in robotics or vehicle crash simulations.
Education: Teaching conservation laws and their applications in classical mechanics.

4. Using the Calculator

Examples:

Example 1: Calculate the final velocities for \(m_1 = 1 \, \text{kg}\), \(v_{1i} = 5 \, \text{m/s}\), \(m_2 = 2 \, \text{kg}\), \(v_{2i} = 0 \, \text{m/s}\), in m/s:
- Enter \(m_1 = 1 \, \text{kg}\), \(v_{1i} = 5 \, \text{m/s}\), \(m_2 = 2 \, \text{kg}\), \(v_{2i} = 0 \, \text{m/s}\).
- Final velocity of object 1: \(v_{1f} = \frac{1 - 2}{1 + 2} \times 5 + \frac{2 \times 2}{1 + 2} \times 0 = \frac{-1}{3} \times 5 = -1.6667 \, \text{m/s}\).
- Final velocity of object 2: \(v_{2f} = \frac{2 \times 1}{1 + 2} \times 5 + \frac{2 - 1}{1 + 2} \times 0 = \frac{2}{3} \times 5 = 3.3333 \, \text{m/s}\).
- Result: \( v_{1f} = -1.6667 \, \text{m/s} \), \( v_{2f} = 3.3333 \, \text{m/s} \).
Example 2: Calculate the final velocities for \(m_1 = 8 \, \text{oz}\), \(v_{1i} = 10 \, \text{ft/s}\), \(m_2 = 1 \, \text{lb}\), \(v_{2i} = -3 \, \text{yd/s}\), in ft/s:
- Enter \(m_1 = 8 \, \text{oz}\), \(v_{1i} = 10 \, \text{ft/s}\), \(m_2 = 1 \, \text{lb}\), \(v_{2i} = -3 \, \text{yd/s}\).
- Convert: \(m_1 = 8 \times 0.028349523125 = 0.226796185 \, \text{kg}\), \(m_2 = 1 \times 0.45359237 = 0.45359237 \, \text{kg}\), \(v_{1i} = 10 \times 0.3048 = 3.048 \, \text{m/s}\), \(v_{2i} = -3 \times 0.9144 = -2.7432 \, \text{m/s}\).
- Final velocity of object 1 (in m/s): \(v_{1f} = \frac{0.226796185 - 0.45359237}{0.226796185 + 0.45359237} \times 3.048 + \frac{2 \times 0.45359237}{0.226796185 + 0.45359237} \times (-2.7432) = \frac{-0.226796185}{0.680388555} \times 3.048 + \frac{0.90718474}{0.680388555} \times (-2.7432) = -1.0160 - 3.6583 = -4.6743 \, \text{m/s}\).
- Final velocity of object 2 (in m/s): \(v_{2f} = \frac{2 \times 0.226796185}{0.680388555} \times 3.048 + \frac{0.45359237 - 0.226796185}{0.680388555} \times (-2.7432) = 2.0320 - 0.9146 = 1.1174 \, \text{m/s}\).
- Convert to ft/s: \(v_{1f} = -4.6743 \times \frac{1}{0.3048} \approx -15.3356 \, \text{ft/s}\), \(v_{2f} = 1.1174 \times \frac{1}{0.3048} \approx 3.6654 \, \text{ft/s}\).
- Result: \( v_{1f} = -15.3356 \, \text{ft/s} \), \( v_{2f} = 3.6654 \, \text{ft/s} \).

5. Frequently Asked Questions (FAQ)

Q: What is an elastic collision?
A: An elastic collision is a collision where both momentum and kinetic energy are conserved, meaning no energy is lost to heat, sound, or deformation.

Q: Why must masses be greater than zero?
A: Zero or negative mass is physically impossible, and the formulas require non-zero masses to avoid division by zero.

Q: What does a negative final velocity mean?
A: A negative final velocity indicates that the object is moving in the opposite direction after the collision, depending on the chosen positive direction.