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.

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  • 1 decade ago
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    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

  • 1 decade ago

    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.

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