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Elastic Collision in Two Dimensions Using Impact Parameter

An elastic collision is an encounter between two bodies in which the total kinetic energy of the two bodies after the encounter is equal to their total kinetic energy before the encounter. Perfectly elastic collisions occur only if there is no net conversion of kinetic energy into other forms.

The final speeds of two objects depend directly on the initial speed between the objects. The masses, the radii of the object and impact parameter also adjust this.

Formula

{QuantityVariable[Subscript["v", "1", "f"], "Speed"] == (Sqrt[QuantityVariable[Subscript["m", "1"], "Mass"]^2 - 2*Cos[2*ArcSin[QuantityVariable["b", "Length"]/(QuantityVariable[Subscript["r", "1"], "Radius"] + QuantityVariable[Subscript["r", "2"], "Radius"])]]*QuantityVariable[Subscript["m", "1"], "Mass"]*QuantityVariable[Subscript["m", "2"], "Mass"] + QuantityVariable[Subscript["m", "2"], "Mass"]^2]*QuantityVariable[Subscript["v", "1", "i"], "Speed"])/(QuantityVariable[Subscript["m", "1"], "Mass"] + QuantityVariable[Subscript["m", "2"], "Mass"]), QuantityVariable[Subscript["v", "2", "f"], "Speed"] == (2*QuantityVariable[Subscript["m", "1"], "Mass"]*Sqrt[1 - QuantityVariable["b", "Length"]^2/(QuantityVariable[Subscript["r", "1"], "Radius"] + QuantityVariable[Subscript["r", "2"], "Radius"])^2]*QuantityVariable[Subscript["v", "1", "i"], "Speed"])/(QuantityVariable[Subscript["m", "1"], "Mass"] + QuantityVariable[Subscript["m", "2"], "Mass"]), QuantityVariable[Subscript["θ", "1", "f"], "Angle"] == ArcTan[Quantity[1, "Kilograms"^(-1)]*(QuantityVariable[Subscript["m", "1"], "Mass"] - Cos[2*ArcSin[QuantityVariable["b", "Length"]/(QuantityVariable[Subscript["r", "1"], "Radius"] + QuantityVariable[Subscript["r", "2"], "Radius"])]]*QuantityVariable[Subscript["m", "2"], "Mass"]), Quantity[1, "Kilograms"^(-1)]*QuantityVariable[Subscript["m", "2"], "Mass"]*Sin[2*ArcSin[QuantityVariable["b", "Length"]/(QuantityVariable[Subscript["r", "1"], "Radius"] + QuantityVariable[Subscript["r", "2"], "Radius"])]]], QuantityVariable[Subscript["θ", "2", "f"], "Angle"] == ArcSin[QuantityVariable["b", "Length"]/(QuantityVariable[Subscript["r", "1"], "Radius"] + QuantityVariable[Subscript["r", "2"], "Radius"])]}

symbol description physical quantity
v1,f final speed 1 "Speed"
m1 mass 1 "Mass"
m2 mass 2 "Mass"
b impact parameter "Length"
r1 radius 1 "Radius"
r2 radius 2 "Radius"
v1,i initial speed 1 "Speed"
v2,f final speed 2 "Speed"
θ1,f final angle 1 "Angle"
θ2,f final angle 2 "Angle"

Forms

Examples

Get the resource:

In[1]:=
ResourceObject["Elastic Collision in Two Dimensions Using Impact \ Parameter"]
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Get the formula:

In[2]:=
FormulaData[ ResourceObject[ "Elastic Collision in Two Dimensions Using Impact Parameter"]]
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Use some values:

In[3]:=
FormulaData[ ResourceObject[ "Elastic Collision in Two Dimensions Using Impact Parameter"], \ {QuantityVariable[ \!\(\*SubscriptBox[\("v"\), \("1", "i"\)]\),"Speed"] -> Quantity[1, ("Meters")/("Seconds")], QuantityVariable[ \!\(\*SubscriptBox[\("r"\), \("1"\)]\),"Radius"] -> Quantity[0.057`, "Meters"]}]
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