Elastic Collisions in Two Dimensions

Two small hovercrafts show

Elastic Collisions in two dimensions.

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Theory:

     This demo shows the law of conservation of momentum. This law states that the total momentum, P, of an isolated system stays constant, regardless of happens inside the system. This can be expressed mathematically as:

P_i = P_f

where P_i is the initial momentum of the system and P_f is the final momentum.

The collisions in this demo are elastic. This means that the two hovercrafts will collide with each other, but they will not stick together afterwords. As the momentum of an object is just their mass times their velocity, the conservation of momentum can be expressed as:

mv_1i + mv_2i = mv_1f + mv_2f

where m is the mass of a hovercraft (here we are assuming the two hovercrafts have the same mass), v_1i is the initial velocity of the first hovercraft, v_2i is the initial velocity of the second hovercraft, v_1f is the final velocity of the first hovercraft and v_2f is the final velocity of the second hovercraft.

However, because the collision is in two dimensions, this equation needs to be split into its x and y components. Thus, the equations are:

x:   mv_1xi + mv_2xi = mv_1xf + mv_2xf 

y:   mv_1yi + mv_2yi = mv_1yf + mv_2yf 

It is also useful to keep in mind that during a perfectly elastic collision, the mechanical energy is conserved as well. This means that:

K_i = K_f

where Ki is the initial kinetic energy and Kf is the final kinetic energy. Splitting this into its x and y components, we get:

x:   (1/2) m (v_1xi)² + (1/2) m (v_2xi)² = (1/2) m (v_1xf)² + (1/2) m (v_2xf)²

y:   (1/2) m (v_1yi)² + (1/2) m (v_2yi)² = (1/2) m (v_1yf)² + (1/2) m (v_2yf)²

Both the energy and momentum equations can be used to analyze elastic collisions in two dimensions. 

Apparatus:

• Two small hovercrafts

Procedure:

• Turn on the hovercrafts and place them on a flat area for them to move around on.
• Try sliding the hovercrafts into each other in various ways, like in the video. Observe the speeds of the hovercrafts before and after the collisions.