# On an air track, a 400g glider?

On an air track, a 400 g glider moving to the right at 2.00 m/s collides elastically with a 500 g glider moving in the opposite direction at 3.00 m/s.

Find the velocity of first glider after the collision.

Find the velocity of second glider after the collision.

Relevance

Nomenclature:

Smaller glider mass: m1

Larger glider mass: m2

Initial velocity of smaller glider: v1i

Initial velocity of larger glider: v2i

Final velocity of smaller glider: v1f

Final velocity of larger glider: v2f

Replacement of v with u is for the center of mass reference frame velocities. v is laboratory reference frame.

Coordinate system: right is positive, left is negative. All velocities comply with this sign convention, thus v2i is actually a negative number.

The velocities you gave are in the laboratory reference frame. Nothing is special about the laboratory reference frame other than that it is what we use to take our measurements.

A nifty shortcut occurs when you look at this from the center of mass reference frame.

What is the velocity of the center of mass?

vcm = (m1*v1i + m2*v2i)/(m1 + m2)

In order to satisfy conservation of momentum, the velocity of the center of mass must remain the same before and after.

In the center of mass reference frame, initial velocities are:

u1i = v1i - vcm

u2i = v2i - vcm

In order to satisfy both conservation of mass and conservation of kinetic energy (condition of elastic collision), the bodies simply switch direction in the center of mass reference frame.

u1f = -u1i

u2f = -u2i

Thus:

u1f = vcm - v1i

u2f = vcm - v2i

Translate back to the laboratory reference frame:

v1f = u1f + vcm

v2f = u2f + vcm

Substitute:

v1f = 2*vcm - v1i

v2f = 2*vcm - v2i

And recall value of vcm:

v1f = 2*(m1*v1i + m2*v2i)/(m1 + m2) - v1i

v2f = 2*(m1*v1i + m2*v2i)/(m1 + m2) - v2i

Simplify and get concluding expressions:

v1f = (m1*v1i + 2*m2*v2i - v1i*m2)/(m1+m2)

v2f = (m2*v2i + 2*m1*v1i - v2i*m1)/(m1+m2)

Data:

m1:= 400 grams; m2:=500 grams; v1i:=+2 m/s; v2i:= -3 m/s;

Results:

v1f = -3.556 m/s, negative indicates toward the left

v2f = +1.444 m/s, positive indicates toward the right

• Calculate the initial momentum (mass*velocity) of each glider. After the collision, the first glider will have a momentum equal to the initial momentum of the second, and second will have a momentum equal the initial momentum of the first. Divide momentum by mass to find velocity for each glider.