The three air carts shown in the figure(Figure 1) have masses, reading from left

Question

The three air carts shown in the figure(Figure 1) have masses, reading from left to right, of 4m, 2m, and m, respectively. The most massive cart has an initial speed of v0; the other two carts are at rest initially. All carts are equipped with spring bumpers that give elastic collisions. (Assume the air track is long enough to accommodate all collisions.)

Find the final speed of the cart with mass 2m. Express your answer in terms of v0.

Find the final speed of the cart with mass m. Express your answer in terms of v0.

Explanation / Answer

the initial momentum of all of the masses

= 4mv0

now

after the collission the final momentum is

4 m vf + ( 2 m + m )U

here U = is the resultant velocity of 2m and m

4 m vf + 3 mU = 4mv0  

4 ( v0 - v f ) = 3U ---- 1

now

taking the kinematic energy is

4 m ( v02 - vf2 ) = 3 m U2

4 ( v0 + vf ) ( v0 - vf ) = 3 U2 -----2

( v0 + vf ) X 3 U = 3 U2

now by susbstituting

4 ( v0 - vf ) = 3 U

( v0 + vf ) = U ----- 3

now equations 1 and 2

U = 1.4312 v0  

just considering

the new velocity of 2m is say v

and

the new velocity of m is say w

then rewriting the equation

2 m X v + m X w = 0
2 X v + w = 0

but v + w = U

then

v = - U

so

v = - 1.4312 v0  

now for w

w = - 2 X v

w = - 2.8624 v0

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