Visualize 1D and 2D collisions in real time. Select elastic, inelastic, or coefficient of restitution e collision types and verify momentum conservation and kinetic energy change numerically.
Parameters
Dimension Mode
CollisionType
Coefficient of restitution e
Presets
Mass m₁
kg
Mass m₂
kg
Initial velocity v₁
m/s
Initial velocity v₂
m/s
Angle θ₁
°
Angle θ₂
°
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Kinetic Energy Before [J]
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Kinetic Energy After [J]
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Energyloss [J]
Results
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v₁' After Collision [m/s]
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v₂' After Collision [m/s]
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Momentum Before [kg·m/s]
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Momentum After [kg·m/s]
Visualization
Drag objects to set initial velocity vectors (arrow direction = velocity direction)
t = 0.000 s
#
e
v₁'
v₂'
ΔKE [J]
Theory & Key Formulas
Law of conservation of momentum (regardless of collision type):
$$m_1v_1 + m_2v_2 = m_1v_1' + m_2v_2'$$
Definition of coefficient of restitution: $e = -\dfrac{v_1'-v_2'}{v_1-v_2}$ (elastic: $e=1$, perfectly inelastic: $e=0$)
What exactly is the "coefficient of restitution" (e) in this simulator? I see it's a slider from 0 to 1.
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Basically, it's a number that tells you how "bouncy" a collision is. In practice, e=1 means a perfectly elastic collision where no kinetic energy is lost. e=0 means a perfectly inelastic collision where the objects stick together. Try moving the slider above and watch how the final velocities and the energy loss chart change instantly.
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Wait, really? So if momentum is always conserved, what's the difference between elastic and inelastic? Is it just about energy?
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Exactly! The law of conservation of momentum holds true for any collision, as long as there's no external force. The key difference is kinetic energy. For instance, in a car crash test (a very inelastic event, e ≈ 0), momentum is conserved between the car and barrier, but a huge amount of kinetic energy is transformed into sound, heat, and deformation. In this simulator, you can verify this by setting e=1 and then e=0 while keeping the same initial velocities.
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That makes sense. But this simulator also has angles (θ). How does momentum conservation work in 2D?
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Great question! In two dimensions, momentum is conserved independently in the x and y directions. It's like having two separate 1D collisions happening at right angles. A common case is a glancing blow in billiards. When you change the angle parameters θ₁ and θ₂ in the simulator, you're defining the direction of each object's velocity vector. The tool then calculates the momentum components in each direction before and after the collision to show you the vector nature of conservation.
Physical Model & Key Equations
The fundamental rule for any collision is the conservation of total momentum. This holds true regardless of whether the collision is elastic or inelastic. For two objects colliding in one dimension, the equation is:
$$m_1v_1 + m_2v_2 = m_1v_1' + m_2v_2'$$
Here, $m_1$, $m_2$ are the masses, $v_1$, $v_2$ are the initial velocities, and $v_1'$, $v_2'$ are the final velocities after the collision.
The coefficient of restitution (e) defines the "bounciness" and relates the relative speed of separation to the relative speed of approach. It determines how much kinetic energy is lost.
$$e = -\frac{v_1' - v_2'}{v_1 - v_2}$$
$e = 1$: Perfectly Elastic (Kinetic Energy Conserved). $e = 0$: Perfectly Inelastic (Maximum Energy Loss, objects move together). $0 < e < 1$: Real-world inelastic collision. Combining this equation with momentum conservation lets you solve for the final velocities.
Frequently Asked Questions
1D collisions simulate collisions along a straight line. In 2D collisions, velocity vectors and angles are also visualized when objects collide obliquely, allowing you to observe more realistic collision phenomena. You can easily try this by switching modes.
e=0.5 corresponds to an inelastic collision, where part of the kinetic energy is lost after the collision. You can compare the kinetic energy before and after the collision using the numerical display, confirming that the lost energy has been converted into heat or deformation.
Yes, based on the law of conservation of momentum, calculations are accurate even when the mass difference is large. For example, when a light object collides with a heavy object, you can observe in real time how the velocity of the light object changes significantly.
Yes, even while the simulation is running, you can change the mass or velocity vector of each object using sliders or numerical input. Changes are reflected immediately, allowing you to dynamically track changes in momentum and energy.
Real-World Applications
Automotive Crash Testing & Safety Design: Engineers use collision principles to design crumple zones and airbags. By simulating inelastic collisions (low e), they calculate how much kinetic energy must be absorbed by the vehicle's structure to protect passengers, directly informing the design of materials and geometry.
Sports Equipment Design: The coefficient of restitution is critical in designing equipment like golf clubs, tennis rackets, and baseball bats. Designers aim for specific e values to optimize the transfer of momentum and energy from the equipment to the ball, balancing performance with regulations.
Particle Physics & Astrophysics: In particle accelerators, analyzing collisions between subatomic particles relies on conservation laws to deduce properties of new particles. Similarly, astronomers use these principles to model cosmic events like galaxy mergers, where stars "collide" with very high e (nearly elastic).
CAE Software (LS-DYNA, Abaqus) Pre-Analysis: Before running complex and time-consuming finite element simulations, engineers use simple momentum conservation calculations—exactly like this simulator—to estimate post-impact velocities and energy dissipation. This is essential for initial design stages of safety barriers, packaging, and impact absorbers.
Common Misconceptions and Points to Note
First, understand that "the coefficient of restitution e is not a constant". While you set it as a fixed value in the tool, the e of a real object varies with impact velocity, temperature, and material condition. For example, the same rubber ball becomes harder when cold, leading to a higher e, while in a high-speed collision, deformation may not keep up, causing e to decrease. It's risky to set e=0.8 in the tool and conclude "this represents an iron ball." The key is to treat it as a parameter for grasping behavioral trends.
Next, identifying the "collision plane" in 2D collisions. The tool automatically calculates the normal direction of the collision plane, but when considering oblique collisions in practice, defining "which plane is being hit" is the starting point for everything. For instance, if a car hits a wall with a 10-degree incline, the collision plane is the wall's surface (tilted 10 degrees from the ground), and you must not use the horizontal ground velocity directly. Use the simulator to change the angle and observe how the velocity vector decomposes.
Also, it's crucial not to confuse "conservation of momentum" with "force". This tool calculates the states "immediately before" and "immediately after" a collision; it does not output what large forces acted during the collision "instant". To estimate the force (impulse), divide the change in momentum $\Delta p$ by the very short collision time $\Delta t$: $F = \Delta p / \Delta t$. Even in a perfectly inelastic collision (e=0), momentum is conserved, but greater force leads to more severe deformation. Understand that the tool's output is strictly the "result" after the collision.