If one cart was at rest, and was struck by another cart of equal mass, what would the final velocities be for a perfectly elastic collision? For a perfectly inelastic collision?

2 Answers
Nick
Apr 4, 2018

For a perfectly elastic collision, the final velocities of the carts will each be 1/2 the velocity of the initial velocity of the moving cart.

For a perfectly inelastic collision, the final velocity of the cart system will be 1/2 the initial velocity of the moving cart.

Explanation:

For an elastic collision, we use the formula

#m_(1)v_(1i) + m_(2)v_(2i) = m_(1)v_(1f) + m_(2)v_(2f)#

In this scenario, momentum in conserved between the two objects.
In the case where both objects have equal mass, our equation becomes

#m(0) + mv_(0) = mv_(1) + mv_(2)#

We can cancel out m on both sides of the equation to find

#v_(0) = v_1 + v_2#

For a perfectly elastic collision, the final velocities of the carts will each be 1/2 the velocity of the initial velocity of the moving cart.

For inelastic collisions, we use the formula

#m_(1)v_(1i) + m_(2)v_(2i) = (m_(1)+m_2)v_(f)#

By distributing out the #v_f#, and then cancelling out m, we find

#v_2 = 2v_f#

This shows us that the final velocity of the two cart system is 1/2 the velocity of the initial moving cart.

Ananda Dasgupta
Apr 4, 2018

For a perfectly elastic collision, the cart that was initially moving comes to a halt, while the other cart moves with velocity #v# (i.e. the velocities get exchanged.

For a perfectly inelastic collision both carts move with a shared velocity of #v/2#

Explanation:

Momentum conservation leads to

#m_1 v_(1i)+m_2 v_(2i) = m_1 v_(1f)+m_2 v_(2f)#

Since, in this problem #m_1 = m_2 = m#, #v_(1i) = 0# and #v_(2i) = v#, we have

#v = v_(1f)+v_(2f)#

This holds for both elastic and inelastic collision.

Perfectly elastic collision

In a perfectly elastic collision, the relative velocity of separation is the same as that of approach (with a negative sign)
So.

#v_(2f)-v_(1f) = v_(1i)-v_(2i) =-v#

Thus #v_(2f) = 0, v_(2i) = v#

**Perfectly inelastic collision#

For a perfectly inelastic collision, the two bodies stick together, so that

#v_(1f) = v_(2f) = 1/2 (v_(1f) + v_(2f)) = 1/2 v#

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