Two carts both 23kg head on elastic collision. First v=18m/s to right. Second v=27.5 m/s to left. Velocities after collision for both?

Question

Just need the velocities for both after collison. Formula would be nice

 

Answer 1

Elastic collisions are bouncy collisions and the two cars will fly away from each other with different velocities after the collision.
Elastic collisions must satisfy two conditions.

There are Two Simultaneous equations you must solve.
ΣM_initial = ΣM_final
and
KE_initial = KE_final

SO

m1v1 + m2v2 = m1v1_final + m2v2_final

1/2(m1v1²) + 1/2(m2v2²) = 1/2(m1v1_f²) + 1/2(m2v2_f²)

Solve these equations.

Answer 2

Celeb,

 

This is a classic conservation of momentum collision problem. Since the collision is elastic there is no energy lost to friction, car deformation, etc. The carts collide and stick to each other.  The cart that has more momentum will dictate the direction of travel after the collision.

 

Let M₁ & M₂ = masses of the two carts (both identical at 23 kg each)

Let V₁ & V₂ = the velocities of the two cars just before their head-on collision (V₁ = 18 m/s & V₂ = - 27.5 m/s).

Let        V   = velocity of both cars after the collision.

 

The Momentum before the collision is : M₁ V₁ + M₂ V₂ = (23) (18) + (23) (-27.5) = - 218.5 kg m/s

After the collision, the Momentum is:  (M1 + M2) V  = 46 V

 

Set these equal since total system momentum is conserved:  46 V = - 218.5

So  V = - 4.75 m/s to the left.