An air-track cart with mass m = 0.20 kg and speed upsilon_0 = 1.0 m/s approaches

Question

An air-track cart with mass m = 0.20 kg and speed upsilon_0 = 1.0 m/s approaches two other carts that are at rest and have masses 2m and 3m, as indicated in (Figure 1). The carts have bumpers that make all the collisions elastic. Find the final speed of cart 1, assuming the air track extends indefinitely in either direction. Find the final speed of cart 2, assuming the air track extends indefinitely in either direction. Find the final speed of cart 3, assuming the air track extends indefinitely in either direction.

Explanation / Answer

from conservation of linear momentum , we get

m1u1 + m2u2 = m1v1 + m2v2

mx1.0 + 2mx0 = mxv1 + 2mxv2

or, 1 = v1 + 2v2 ..........................1)

from conservation of KE

0.5 m1u1^2 + 0.5m2u2^2 = 0.5m1v1^2 + 0.5m2v2^2

or, mx(1.0)^2 = mv1^2 + 2mv2^2

or, 1 = v1^2 + 2v2^2 ............................2)

solving 1) and 2)

1 = (1-2v2)^2 + 2v2^2

or, 6v2^2 -4v2 = 0

or, v2 (6v2 -4)= 0

or, v2 = 4/6 = 0.67 m/s .....................................ans part b)

putting this in 1)

1 = v1 + 2x4/6

or, v1 = -0.33 m/s [negative sign mean toward left] ...........................ans part a)

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