Vector Bounce Calculator
An advanced tool for accurately calculating bounce using vectors, essential for physics simulations and game development.
Velocity along the horizontal axis (units/sec).
Velocity along the vertical axis (units/sec).
Direction perpendicular to the surface (horizontal).
Direction perpendicular to the surface (vertical).
A value from 0 (no bounce) to 1 (perfect bounce).
18.03 units/s
15.62 units/s
-15.00
27.00
v_out = v_in - (1 + e) * (v_in ⋅ n) * n. This formula calculates the reflection vector where energy loss is controlled by the coefficient of restitution ‘e’.
Bounce Visualization
| Parameter | Value | Unit/Type |
|---|---|---|
| Initial Velocity (Vector) | (10.00, -15.00) | Vector (x, y) |
| Surface Normal (Normalized) | (0.00, 1.00) | Unit Vector |
| Coefficient of Restitution | 0.80 | Scalar |
| Final Velocity (Vector) | (10.00, 12.00) | Vector (x, y) |
What is Calculating Bounce Using Vectors?
Calculating bounce using vectors is a fundamental technique in physics, computer graphics, and game development for simulating how an object reflects off a surface. Instead of simple, one-dimensional bouncing, this method uses vectors to represent the object’s velocity and the surface’s orientation in 2D or 3D space. This allows for realistic simulations of objects hitting surfaces at any angle.
This type of calculation is crucial for anyone creating a physics engine, developing a game with dynamic objects (like a pool game or a platformer), or simulating physical systems. It accurately models the change in direction and speed after an impact, including energy loss. For more info on core physics, see this physics simulation guide.
The Vector Bounce Formula and Explanation
The core of calculating a bounce is the vector reflection formula, which is adapted to include the concept of elasticity (bounciness). The standard formula is:
v_out = v_in - (1 + e) * (v_in ⋅ n) * n
This formula might look complex, but it’s built from simple parts. It takes the initial velocity and subtracts a component that is aligned with the surface normal, effectively “reflecting” it across the plane of the surface.
Formula Variables
| Variable | Meaning | Unit (Auto-inferred) | Typical Range |
|---|---|---|---|
v_in |
The initial velocity vector of the object before impact. | Vector (units/sec) | Any valid 2D/3D vector. |
v_out |
The final (output) velocity vector of the object after impact. | Vector (units/sec) | Calculated result. |
e |
The coefficient of restitution, a scalar representing bounciness. | Unitless ratio | 0.0 to 1.0 |
n |
The normalized (unit length) vector perpendicular to the surface at the point of impact. | Unit Vector | Vector with a magnitude of 1. |
(v_in ⋅ n) |
The dot product of the initial velocity and the surface normal. It measures how much the velocity is directed “into” the surface. | Scalar | Negative for collision. |
Practical Examples of Calculating Bounce
Example 1: Ball Hitting a Horizontal Floor
Imagine a ball moving down and to the right, hitting a flat, level floor. The floor’s normal vector points straight up.
- Inputs:
- Initial Velocity (
v_in):(5, -20)(moving right and strongly down) - Surface Normal (
n):(0, 1)(pointing straight up) - Coefficient of Restitution (
e):0.7(a fairly bouncy ball)
- Initial Velocity (
- Calculation:
- Dot Product:
(5*0 + -20*1) = -20 - Impulse:
-(1 + 0.7) * -20 = 1.7 * 20 = 34 - Final Velocity:
(5, -20) + 34 * (0, 1) = (5, -20) + (0, 34) = (5, 14)
- Dot Product:
- Result: The final velocity is
(5, 14). The horizontal speed is unchanged, while the vertical speed is reversed and slightly reduced due to energy loss. A deep dive on this can be found in our article about elastic collisions.
Example 2: Billiard Ball Hitting a 45-Degree Rail
A ball travels horizontally and strikes a rail angled at 45 degrees. The normal to this surface points inwards, up and to the left.
- Inputs:
- Initial Velocity (
v_in):(15, 0)(moving purely right) - Surface Normal (
n):(-0.707, 0.707)(normalized vector for a 45° angle) - Coefficient of Restitution (
e):0.9(very bouncy)
- Initial Velocity (
- Calculation:
- Dot Product:
(15*-0.707 + 0*0.707) = -10.605 - Impulse:
-(1 + 0.9) * -10.605 = 1.9 * 10.605 = 20.15 - Final Velocity:
(15, 0) + 20.15 * (-0.707, 0.707) = (15, 0) + (-14.24, 14.24) = (0.76, 14.24)
- Dot Product:
- Result: The final velocity is
(0.76, 14.24). The ball is now moving mostly upwards and slightly to the right, as expected from a bank shot.
How to Use This Vector Bounce Calculator
This tool makes calculating bounce using vectors simple. Follow these steps for an accurate result:
- Enter Initial Velocity: Input the X and Y components of your object’s velocity vector before it hits the surface. Positive Y is ‘up’, and positive X is ‘right’.
- Define the Surface: Input the X and Y components of the surface normal vector. This vector must be perpendicular to the surface. For a flat floor, this is (0, 1). For a wall on the right, it’s (-1, 0). The calculator will automatically normalize this vector for you.
- Set the Bounciness: Adjust the ‘Coefficient of Restitution’ slider. A value of 1.0 means no energy is lost (perfectly elastic), while 0.0 means the object stops dead (perfectly inelastic).
- Interpret the Results: The calculator instantly provides the final velocity vector, initial and final speeds, and key intermediate values. The visualization helps you see the interaction spatially. For more advanced topics check out our advanced physics models.
Key Factors That Affect Vector Bounce
The outcome of calculating bounce is determined by three critical factors:
- Impact Angle: The angle between the initial velocity vector and the surface normal is the most critical factor. A direct, head-on collision (velocity is opposite to the normal) will produce a simple reflection, while a glancing blow will result in a significant directional change.
- Coefficient of Restitution (e): This scalar value directly controls the energy preserved in the collision. A higher ‘e’ results in a final speed that is closer to the initial speed, creating a “livelier” bounce. This factor is essential for realistically modeling different materials.
- Initial Speed: While the directional change is determined by the angle and normal, the initial speed scales the entire interaction. A faster object will have a faster final velocity, and the magnitude of the impulse (the force of the bounce) will be greater.
- Surface Normal Orientation: The direction of the normal vector defines the plane of reflection. An incorrect normal will lead to a physically impossible bounce. This is a common bug in game physics, which our debugging game physics guide covers.
- Normalization: The bounce formula requires a “unit normal” (a vector with a length of 1). Our calculator handles this automatically, but in manual calculations, failing to normalize the normal vector will incorrectly scale the resulting bounce velocity.
- Dimensionality: While this calculator operates in 2D, the principles are the same in 3D. The vectors would simply have three components (x, y, z) instead of two, but the vector bounce formula remains identical.
Frequently Asked Questions (FAQ)
What is a “surface normal”?
A surface normal is a vector that points directly outward from a surface, perpendicular (at a 90-degree angle) to that surface at a specific point. It defines the surface’s orientation in space. For a flat floor, the normal is (0, 1), pointing up.
Why are the units ‘units/sec’?
Because this is an abstract calculator for the underlying physics principle, the units are generic. ‘Units’ could be meters, pixels, feet, or any other measure of distance. The calculation is valid regardless of the specific unit system.
What happens if the coefficient of restitution is greater than 1?
A coefficient of restitution greater than 1 represents a collision that generates energy, which is not possible in most classical physics scenarios. It would mean the object bounces back faster than it approached. This can be used in games for special effects, like a “super bouncy” surface.
What if the dot product is positive?
A positive dot product (v_in ⋅ n) means the object is already moving away from the surface, so a collision isn’t actually happening. In a real physics engine, the bounce calculation would be skipped in this case.
Does this calculator work for 3D bounce?
The principle and formula are exactly the same for 3D. You would just use vectors with three components (x, y, z) for the initial velocity and surface normal. The dot product and vector arithmetic extend seamlessly to three dimensions.
Why does my object lose speed even with high restitution?
For any coefficient of restitution less than 1.0, some energy must be lost. This is converted into heat, sound, or deformation of the object. The final speed will therefore be lower than the initial speed. Only a value of 1.0 results in the speeds being equal.
How do I find the normal for a curved surface?
For a curved surface like a circle or sphere, the normal at any point is the vector from the center of the circle to that point on the circumference. For more complex curves, it’s found using calculus (specifically, the gradient).
Can I use this for light reflection?
Yes, the formula for specular reflection of light is a special case of this, where the coefficient of restitution is effectively 1. The formula simplifies to v_out = v_in - 2 * (v_in ⋅ n) * n. Explore more on our ray tracing guide.