2025-08-21 14 min
Collision Calculator
Analyze 1D collisions using conservation of momentum and coefficient of restitution.
Calculator
Mass 1(kg) Help
kg
Must be positive
Velocity 1(m/s)
m/s
Mass 2(kg) Help
kg
Must be positive
Velocity 2(m/s)
m/s
Collision Mode
Decimal Precision
Results update live as you type
Results
Normal
Final Velocity 1
0m/s
Final Velocity 2
10m/s
Impulse
50N*s
KE Before
250J
KE After
250J
KE Lost
0J
% Energy Lost
0%
Visualizations
Results Overview
10m/sFinal Velocity 2
50m/sImpulse
250m/sKE Before
250m/sKE After
Formula
Conservation of linear momentum and coefficient of restitution for 1D two-body collisions.
m1*v1 + m2*v2 = m1*v1' + m2*v2'
Alternative Form
e = (v2' - v1') / (v1 - v2)
Variables
| Symbol | Name | Unit | Description |
|---|---|---|---|
| m1 | Mass 1 | kg | Mass of object 1 |
| v1 | Velocity 1 | m/s | Initial velocity of object 1 |
| m2 | Mass 2 | kg | Mass of object 2 |
| v2 | Velocity 2 | m/s | Initial velocity of object 2 |
| e | Restitution | - | Coefficient of restitution (0 to 1) |
Explanation
Momentum is conserved in all collisions. The coefficient of restitution determines how elastic the collision is: e=1 is perfectly elastic, e=0 is perfectly inelastic.
Derivation
From conservation of momentum and the definition of e, solve the two equations simultaneously for v1_prime and v2_prime.
Step-by-Step Calculation
Follow each step to understand how the result is calculated from your inputs.
Step 1
Enter masses and initial velocities for both objects
Result: Inputs entered
Step 2
Select collision mode (elastic, inelastic, or partially elastic)
Result: Mode selected
Step 3
Apply conservation of momentum: m1v1 + m2v2 = m1v1' + m2v2'
Result: Momentum conserved
Step 4
Apply coefficient of restitution: e = (v2' - v1') / (v1 - v2)
Result: Final velocities computed
Step 5
Calculate impulse J = m1(v1' - v1) and kinetic energy before/after
Result: Results displayed
Final Answer
0
Worked Examples
See how the calculator works with different scenarios.
Scenario
A 5 kg ball at 10 m/s hits a stationary 5 kg ball.
Inputs
mass1:5
velocity1:10
mass2:5
velocity2:0
restitution:1
Calculation
v1' = (5-5)*10/(5+5) = 0 m/s, v2' = 2*5*10/(5+5) = 10 m/s
Result
Balls exchange velocities. KE is fully conserved.
Interpretation
Equal-mass elastic collisions produce complete velocity exchange.
Understanding Your Result
Your Result: Final Velocity 1: 0 m/s
Typical range: Steel on steel: e ~ 0.56. Rubber on concrete: e ~ 0.75. Car crashes: e ~ 0.1-0.2.
What This Means
Collision analysis determines final velocities, impulse, and energy dissipation from two objects impacting each other.
Why It Matters
Essential for vehicle safety design, sports equipment, ballistic analysis, and understanding fundamental mechanics.
Professional Interpretation
Use e values from material testing for accurate predictions. Verify against experimental data for critical designs.
Recommendations
- •Use elastic mode for hard material collisions
- •Use inelastic mode for crash analysis
- •Measure or look up COR values for specific materials
Educational Content
Introduction
Collision analysis is a fundamental topic in classical mechanics that applies conservation of momentum to predict post-impact motion.
Concept Explanation
In every collision, momentum is conserved. The coefficient of restitution determines the fraction of kinetic energy retained. Elastic collisions conserve KE; inelastic collisions dissipate it as heat and deformation.
Applications
▸Vehicle crash analysis
▸Sports equipment design
▸Ballistic pendulum measurements
▸Particle physics
▸Aerospace impact analysis
Benefits
- •Predicts post-collision velocities
- •Quantifies energy dissipation
- •Enables safety design
- •Foundation for multi-body dynamics
Limitations
- •Assumes point masses
- •Ignores rotation and spin
- •1D analysis only
- •Requires known coefficient of restitution
Best Practices
- •Measure e experimentally for your materials
- •Consider both momentum and energy equations
- •Account for rotation in real applications
- •Use safety factors for design
Common Misconceptions
Myth: Momentum is only conserved in elastic collisions
Truth: Momentum is conserved in ALL collisions as long as no external forces act.
Practical Tips
- •e = 1 is perfectly elastic (rare in practice)
- •e = 0 means objects stick together
- •Steel on steel has e ~ 0.56
- •Billiard balls have e ~ 0.96
Common Mistakes
Assuming KE is conserved in all collisions
Only elastic collisions conserve KE. Inelastic collisions always lose KE.
Forgetting to account for direction
Velocities are vectors. Use positive/negative to indicate direction in 1D.
Assumptions
This calculator makes the following assumptions to provide estimates:
Point Masses: Objects are treated as point masses with no rotation.
No External Forces: Gravity and friction are ignored during the brief collision.
Limitations
Be aware of these limitations when using this calculator:
1D Only: This calculator handles head-on collisions only.
No Rotation: Spin effects are not modeled.
Frequently Asked Questions
Answers to common questions about this calculator.
What is a collision in physics?
A collision is an event where two or more bodies exert forces on each other for a relatively short time. Collisions are classified as elastic (kinetic energy conserved), inelastic (kinetic energy lost), or perfectly inelastic (bodies stick together). Momentum is conserved in all types of collisions as long as no external forces act.
What is the law of conservation of momentum?
The total momentum of an isolated system remains constant before and after a collision. For two objects: m1v1 + m2v2 = m1v1' + m2v2', where primed variables are post-collision velocities. This law holds regardless of whether the collision is elastic or inelastic.
What is the coefficient of restitution?
The coefficient of restitution (e) measures the bounciness of a collision. It is defined as e = (v2' - v1') / (v1 - v2), the ratio of relative separation velocity to relative approach velocity. e = 1 for perfectly elastic, e = 0 for perfectly inelastic, and 0 < e < 1 for partially elastic collisions.
What is an elastic collision?
In an elastic collision, both momentum and kinetic energy are conserved. The objects bounce off each other without any energy loss to heat, sound, or deformation. Billiard ball collisions and atomic/particle collisions are approximately elastic. For equal masses with one initially at rest, the objects exchange velocities.
What is an inelastic collision?
In an inelastic collision, momentum is conserved but kinetic energy is not. Some kinetic energy is converted to heat, sound, or deformation energy. Car crashes, where metal crumples and energy is absorbed, are examples of inelastic collisions.
What is a perfectly inelastic collision?
A perfectly inelastic collision occurs when the colliding objects stick together after impact, moving with a common velocity. This is the maximum energy loss possible while still conserving momentum. The common velocity is v' = (m1v1 + m2v2) / (m1 + m2).
What is impulse in a collision?
Impulse (J) is the change in momentum of an object: J = delta_p = m * delta_v = F * delta_t. The impulse-momentum theorem states that the impulse applied to an object equals its change in momentum. Impulse has units of N*s or kg*m/s.
How is kinetic energy lost in inelastic collisions?
The kinetic energy lost is converted to internal energy (heat), deformation energy, sound energy, and electromagnetic radiation. For a perfectly inelastic collision, the fraction of KE lost is m2/(m1+m2) when object 2 is initially at rest.
What is the center of mass frame in collisions?
The center-of-mass (CM) frame is a reference frame where the total momentum is zero. In this frame, both objects approach the CM and leave it with equal and opposite momenta. Collisions appear symmetric in the CM frame, making analysis simpler.
How do car crashes demonstrate collision physics?
Car crashes are highly inelastic (e approximately 0.1-0.2). The front crumple zone absorbs kinetic energy through metal deformation, reducing peak forces on passengers. Seatbelts and airbags extend the deceleration time, reducing the impulse force.
What happens in a collision between unequal masses?
When a heavy object strikes a lighter one, the lighter object gains more velocity. For example, a bowling ball (7 kg) hitting a ping pong ball (0.003 kg) sends the ball flying at roughly 23 times the bowling ball speed.
Why do billiard balls exchange velocities?
When two billiard balls of equal mass collide elastically with one at rest, the moving ball stops and the stationary ball moves off with the original velocity in a head-on collision. At angles, they depart at 90 degrees to each other.
What is a ballistic pendulum?
A ballistic pendulum measures projectile velocity using a perfectly inelastic collision followed by energy conservation. A bullet embeds in a suspended block; the block+bullet swings up to height h. Velocity is found from v = ((m+M)/m) * sqrt(2gh).
How do explosions relate to collisions?
An explosion is like a reverse perfectly inelastic collision. Internal energy is converted to kinetic energy, pushing fragments apart. Momentum is conserved: if two fragments of masses m1 and m2 are created, m1v1 = m2v2 for an initially stationary system.
What is the impulse-momentum theorem?
The impulse-momentum theorem states: J = F_avg * delta_t = m * v_f - m * v_i = delta_p. The average force during a collision equals the change in momentum divided by the contact time. This is why longer collision times reduce peak forces on passengers.
How does collision relate to safety engineering?
Safety engineers design crumple zones, airbags, and seatbelts to manage collision energy. The goal is to extend the deceleration time (reducing peak force) while absorbing kinetic energy through controlled deformation.
What is a head-on collision?
In a head-on collision, two objects approach each other directly along the same line. The relative velocity is v1 + v2 (sum of speeds). Head-on collisions are the most severe type because the relative velocity and energy dissipation is maximized.
What happens in a perfectly elastic collision with equal masses?
When two objects of equal mass collide elastically, they exchange velocities. If object 1 is moving at v and object 2 is at rest, after collision object 1 stops and object 2 moves at v. This is the basis for Newton Cradle.
What is a glancing collision?
A glancing collision occurs when objects do not strike head-on but at an angle. In elastic glancing collisions between equal masses (one at rest), the objects depart at exactly 90 degrees to each other.
How do collisions work in particle physics?
In particle physics, collisions at high energies create new particles. The center-of-mass frame is used for analysis. Conservation of momentum, energy, charge, and other quantum numbers govern which reactions are possible.
What is the difference between collision and impact?
Impact generally refers to a single collision event, while collision can describe the entire interaction. Impact force is the instantaneous force during contact. Impact energy is the kinetic energy available for dissipation.
What is a partially elastic collision?
In a partially elastic collision (0 < e < 1), some kinetic energy is conserved but some is lost. Most real-world collisions are partially elastic. A basketball (e ~ 0.8), tennis ball (e ~ 0.75), and golf ball (e ~ 0.83) all lose some energy per bounce.
What is the coefficient of restitution for common materials?
Typical values: steel on steel e = 0.56, rubber on concrete e = 0.75, glass on glass e = 0.65, golf ball on clubface e = 0.83, tennis ball on racket e = 0.75, superball e = 0.9. Values depend on surface conditions and temperature.
How do collisions affect vehicle design?
Vehicle design incorporates crumple zones (progressive deformation), load paths (directing forces around passenger compartment), passenger cell (rigid survival space), restraint systems (seatbelts, airbags), and side impact protection.
What is KE loss in a perfectly inelastic collision?
For a perfectly inelastic collision where object 2 is at rest, KE_loss = 0.5 * m1 * v1^2 * m2/(m1+m2). The lighter the stationary object relative to the moving one, the smaller the fraction of energy lost.
How do collisions work in 2D?
In 2D collisions, momentum is conserved independently in each direction (x and y). For elastic collisions with one object initially at rest and equal masses, the two objects depart at 90 degrees to each other.
What is a collision in sports?
Sports collisions involve complex biomechanics. A baseball bat hitting a ball (e ~ 0.5), soccer ball kicked (e ~ 0.5), golf ball on a driver (e ~ 0.83). Equipment design optimizes the coefficient of restitution and impulse transfer.
How is collision force estimated?
Average collision force F = delta_p / delta_t. For a 1000 kg car at 20 m/s stopping in 0.1 s, F = 2,000,000 N (about 200 g). Increasing stopping time to 1 s reduces force to 200,000 N (about 20 g).
What is the coefficient of restitution for a golf ball?
A golf ball on a driver face has e approximately 0.83. Combined with the driver clubhead speed, this produces ball speeds exceeding 160 mph. The USGA limits COR to 0.83 to maintain equipment standards.
How do collisions relate to Newton Cradle?
Newton Cradle demonstrates momentum and energy conservation through a chain of elastic collisions. A swinging ball transfers momentum through the stationary balls to launch the ball on the opposite end. All collisions occur nearly simultaneously.
What is the work-energy theorem for collisions?
The work-energy theorem states that net work done on an object equals its change in kinetic energy: W_net = delta_KE. In inelastic collisions, some work goes into deformation and heat rather than kinetic energy.
How is momentum conserved in rocket propulsion?
Rocket propulsion uses conservation of momentum: as exhaust gases are expelled backward at high velocity, the rocket moves forward. The total momentum of rocket+exhaust remains constant (in the absence of external forces).
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Glossary
Technical terms used in mortgage calculations explained in plain language.
- Momentum
- The product of mass and velocity (p = mv). A conserved vector quantity.
- Coefficient of Restitution
- Ratio of separation to approach velocity. e = 1 is elastic, e = 0 is perfectly inelastic.
- Impulse
- Change in momentum (J = delta_p). Equals average force times contact time.
- Elastic Collision
- A collision where both momentum and kinetic energy are conserved.
- Inelastic Collision
- A collision where momentum is conserved but kinetic energy is not.
Authoritative References
Trusted sources and standards referenced by this calculator.
Applicable Standards
ISO 16943ASTM E303
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Page Information
Last Updated:
2025-08-21
Version:
2.0.0
License:
Free to use for educational and professional purposes.