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If we push one cart (Cart A) into a second stationary cart (Cart B), v_Ai is nonzero and v_Bi=0, respectively. Assume no outside forces will be involved and the collision is elastic (i.e. both momentum and kinetic energy are conserved). Derive the general equation for the velocity of Cart A after the collision v_Af as well as the velocity of Cart B after the collision v_Bf. We want these to be in terms of known quantities: initial velocity v_Ai and the mass of the carts m_A and m_B. Refer to the "Useful Theory" section below to get you started. If you get stuck, work with your lab partners and refer to your text to find the solution before

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Answer:

[tex]v_A=(\frac{m_A-m_B}{m_A+m_B})u_A[/tex]

[tex]v_B=(\frac{2m_A}{m_A+m_B})u_A[/tex]

Explanation:

In an elastic collision, the total momentum of the system and the total kinetic energy of the system are conserved.

So:

- Conservation of momentum:

[tex]m_A u_A+m_B u_B = m_A v_A + m_B v_B[/tex] (1)

where

[tex]m_A, m_B[/tex] are the masses of cart A and B

[tex]u_A,u_B[/tex] are the initial velocities of cart A and B

[tex]v_A,v_B[/tex] are the final velocities of cart A and B

- Conservation of kinetic energy:

[tex]\frac{1}{2}m_A u_A^2+\frac{1}{2}m_B u_B^2= \frac{1}{2}m_A v_A^2 +\frac{1}{2} m_B v_B^2[/tex] (2)

From (1) we can write:

[tex]m_A (u_A-v_A)=m_B(v_B-u_B)\\\frac{m_A (u_A-v_A)}{m_B(v_B-u_B)}=1[/tex] (3)

From (2) we get:

[tex]m_A (u_A^2-v_A^2)=m_B(v_B^2-u_B^2)\\m_A(u_A-v_A)(u_A+v_A)=m_B(v_B-u_B)(v_B+u_B)\\\frac{m_A(u_A-v_A)}{m_B(v_B-u_B)}(u_A+v_A)=(v_B+u_B)[/tex] (4)

Substituting (3) into (4),

[tex]1\cdot (u_A+v_A)=(v_B+u_B)\\v_B=u_A+v_A-u_B[/tex] (5)

And substituting (5) into (1),

[tex]m_A(u_A-v_A)=m_B (u_A+v_A-2u_B)[/tex]

And now we can solve to find the final expressions:

[tex]v_A=(\frac{m_A-m_B}{m_A+m_B})u_A+(\frac{2m_B}{m_A+m_B})u_B[/tex]

[tex]v_B=(\frac{2m_A}{m_A+m_B})u_A+(\frac{m_B-m_A}{m_A+m_B})u_B[/tex]

But in this problem

[tex]u_B=0[/tex]

So the equations can be rewritten as

[tex]v_A=(\frac{m_A-m_B}{m_A+m_B})u_A[/tex]

[tex]v_B=(\frac{2m_A}{m_A+m_B})u_A[/tex]

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