Energy dissipated in 3-cars collision

In summary, the conversation discusses the concept of inelastic collisions and how to calculate the fraction of initial energy lost in a multi-car collision scenario. The solution involves using momentum and energy conservation principles, and a simpler recursive approach is suggested for solving general cases.
  • #1
geoffrey159
535
72

Homework Statement



Cars B and C are at rest with their brakes off. Car A plows into B at high speed, pushing B into C. If the collisions are completely inelastic, what fraction of the initial energy is dissipated when car
C is struck? The cars are identical initially.

Homework Equations


Collisions

The Attempt at a Solution



First collision:

Momentum is conserved and the collision is inelastic so ##mv_1 = 2mv_2##. The energy dissipated after the first collision
is ## Q_1 = \triangle K = \frac{1}{2} mv_1 ^2 - m v_2^2 = \frac{1}{4}mv_1^2 = \frac{1}{2} K_i ##

Second collision:
Momentum is conserved and the collision is inelastic so ##2mv_2 = 3mv_3##. The energy dissipated after the second collision is ## Q_2 = \triangle K = m v_2 ^2 - \frac{3}{2}m v_3^2 = \frac{1}{3}mv_2^2 = \frac{1}{12}mv_1^2 = \frac{1}{6} K_i ##

Total energy lost is ## Q = Q_1 + Q_2 = \frac{2}{3} K_i ## so the fraction of initial energy dissipated after C is struck is 2/3.

Is that correct?
 
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  • #2
Looks good. Anything else?
 
  • #3
Thank you ! What do you mean by 'anything else ?' ?
 
  • #4
You've not done anything wrong but can you see a simpler way? What if there were 10 cars?
 
  • #5
Yes, there is a recursive way to do it:

With momentum conservation and inelasticity: ## mnv_n = m(n+1) v_{n+1} ##

So ## v_n =\frac{v_1}{n} ## and total energy lost for n-car collision is ## Q = K_i - K_n = (1 - \frac{1}{n}) K_i ##
 
  • #6
Yes, you need only consider energy and momentum at the beginning and end. Even if the cars were different masses.
 
  • #7
I see, it would have been simpler to consider the general case ! Thanks !
 

1. What is energy dissipation in a 3-cars collision?

Energy dissipation in a 3-cars collision refers to the loss of energy that occurs during the process of the collision. This can be due to factors such as deformation of the vehicles, sound and heat production, and friction between the vehicles and the road.

2. How is energy dissipated in a 3-cars collision?

Energy is dissipated in a 3-cars collision through various mechanisms, including the deformation of the vehicles involved, the production of sound and heat, and the transfer of kinetic energy to other forms of energy such as potential energy or thermal energy.

3. What factors affect the amount of energy dissipated in a 3-cars collision?

The amount of energy dissipated in a 3-cars collision can be affected by factors such as the speed and mass of the vehicles involved, the direction of the collision, the type of materials the vehicles are made of, and the condition of the road surface.

4. How is the energy dissipated in a 3-cars collision related to the severity of the collision?

The amount of energy dissipated in a 3-cars collision is directly related to the severity of the collision. A higher amount of energy dissipated typically indicates a more severe collision, which can result in more damage to the vehicles and potential injuries to the passengers.

5. What are some real-world applications of studying energy dissipation in 3-cars collisions?

Studying energy dissipation in 3-cars collisions can have practical applications in improving vehicle safety and designing safer roads and highways. It can also help in understanding the impact of different factors on the severity of collisions and developing measures to reduce the amount of energy dissipated in accidents.

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