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This simulation shows two blocks moving along a track and colliding with each other and the walls. One spring is attached to the wall with a spring. Try changing the mass of the blocks to see if the collisions happen correctly.

You can change parameters such as mass, spring stiffness, and friction (damping). You can drag either block with your mouse to change the starting positions.

The math behind the simulation is shown below. Also available: source code, documentation and how to customize.

Collisions and Conservation of Momentum

The spring and block on the left use the same model as the single spring simulation. For a collision with the wall, we simply reverse the velocity. For collisions between moving blocks we use the law of conservation of momentum to determine the new velocities.

Define the following variables:

v_1i, v_2i = initial (pre-collision) velocity of blocks
v_1f, v_2f = final (post-collision) velocity of blocks
v_cm = velocity of center of mass
v_1cm, v_2cm = velocity of blocks in center of mass frame
m₁, m₂ = mass of blocks
p_i, p_f = initial and final momentum of system of blocks

We find the velocity of the center of mass of the 2-block system.

v_cm =	m₁ v_1i + m₂ v_2i
	m₁ + m₂

Next we find the velocity of each block in the coordinate system (frame) that is moving along with the center of mass.

v_1cm = v_1i − v_cm
v_2cm = v_2i − v_cm

Next we reflect (reverse) each velocity in this center of mass frame, and translate back to the stationary coordinate system.

v_1f = −(v_1i − v_cm) + v_cm
v_2f = −(v_2i − v_cm) + v_cm

If you fully expand the above you get

v_1f = −v_1i +	2(m₁ v_1i + m₂ v_2i)
	m₁ + m₂

(1)

v_2f = −v_2i +	2(m₁ v_1i + m₂ v_2i)
	m₁ + m₂

(2)

As a check, we can calculate the pre- and post- collision momentum, which should be the same.

p_i = m₁ v_1i + m₂ v_2i
p_f = m₁ v_1f + m₂ v_2f

If you expand p_f using equations (1) and (2) and simplify you will see that p_f = p_i as expected.

Handling Collisions in Software

simulation of two balls colliding

Simulations on the computer are discrete in the sense that time advances in "chunks", not smoothly. In the diagram, we calculated the state of the world at time t = 10.0 and then at time t = 10.1 . But the collision happened sometime inbetween. So by the time we detect a collision, the objects are overlapping each other -- a physically impossible state.

The simulations on this website handle collisions as follows:

Detect that a collision occurred
Re-run the simulation to very close to (but just before) the time of collision. The time of collision is found by a binary search process.
Handle the collision by modifying the state of the simulation, such as fixing the velocities as described above.
Continue to run the simulation up to the current 'now' time. If additional collisions happen they are handled in the same way.

If you are not running the simulation in real time, you can take as long as you want to get as much accuracy as desired. But for a real time simulation, you may need to accept lower accuracy or use some fancier programming. For example, instead of using trial and error to find the time of collision, you could try to predict the time of collision based on the state of the system.

Other examples of simulations with collision handling are Roller Coaster with Flight, Molecule 2 and Rigid Body Collisions.

This web page was first published April 2001.

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