# Cars colliding into each other.

• kraigandrews
In summary: You already had an expression for the final kinetic energy in terms of Vo and the masses. Erase all of your work and start again. You've made it way too complicated.Now, the kinetic energy of the A+B pair is (1/2)(MA+MB)*Vo'^2. The kinetic energy of the entire A+B+C system is (1/2)(MA+MB+MC)*Vo''^2. What is the ratio of these two energies?In summary, one can calculate the fraction of initial energy dissipated in the collision involving car C by treating the problem as two separate collisions and using conservation of momentum and the equation for kinetic energy. By finding the velocity of the car A
kraigandrews

## 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. Assume the collisions are completely inelastic. What fraction of the initial energy is dissipated in the collision involving car C?
Data: MA = 1010 kg; MB = 1250 kg; MC = 1470 kg.

Pinitial=Pfinal
KE=.5mv^2

## The Attempt at a Solution

Pi=Pf
m_a*Vo=(m_a+m_b+m_c)*Vo'
Q+.5m_aVo^2=.5(m_a+m_b+m_c)Vo'^2; where Q is the kinetic energy lost

I'm pretty sure these equations are correct but I don't know where to go from here?

Treat the problem as two separate collisions. Prior to any collision, car A has all the available kinetic energy. A collides with B and they stick together and move with a new velocity with momentum is conserved. The pair then have a collision with C, resulting in another new velocity for the trio (KE final).

Ok, so how would I be able to find the fraction of the first one?

kraigandrews said:
Ok, so how would I be able to find the fraction of the first one?

Find the KE after the first collision, then the remaining KE after the second collision. The difference between these two is the KE lost in the second collision. Compare this with what you started with before any collisions.

Ok sorry, I'm struggling with this one. So I have:
for the first collision:
MA*Vo=(MA+MB)*Vo'
then I solved for Vo'=((MA/(MA+MB))Vo

then plugged it into:
.5MAVo^2=.5(MA+MB)Vo'^2=.5((MA^2)/(MA+MB))-Q; I assumed Vo=1
then solved for Q, but then I am unsure of:
1. if this makes any sense
2. if .5MAVo^2=Eo then going forth from here can I do the same for the second collision solving for Q2 then add them together and end up with:
(Eo-(Q+Q2))/Eo for the fraction?

kraigandrews said:
Ok sorry, I'm struggling with this one. So I have:
for the first collision:
MA*Vo=(MA+MB)*Vo'
then I solved for Vo'=((MA/(MA+MB))Vo

then plugged it into:
.5MAVo^2=.5(MA+MB)Vo'^2=.5((MA^2)/(MA+MB))-Q; I assumed Vo=1
then solved for Q, but then I am unsure of:
1. if this makes any sense
2. if .5MAVo^2=Eo then going forth from here can I do the same for the second collision solving for Q2 then add them together and end up with:
(Eo-(Q+Q2))/Eo for the fraction?

Kinetic energy isn't conserved in an inelastic collision. So don't try to write conservation of kinetic energy equations

You found the velocity Vo' of the car A+B tangle, and that will yield the kinetic energy of the pair as it moves towards car C. Also, there's no need to assume that Vo=1. Just keep Vo as Vo... it'll cancel out in the end when you form your ratio of energies.

The KE of cars A+B is (1/2)(MA+MB)*Vo'^2. You can plug in the expression for Vo'.

You've got Vo' and the mass of the A+B pair, so determine the final velocity after the A+B+C collision using conservation of momentum. Use that to find the expression for the final KE.

I'm not sure I follow because if i use consveration of momentum again ill just end up with another variable Vo'' because if i do:
(Ma+Mb)Vo'=(Ma+Mb+Mc)Vo'' it doesn't work out.
so i am confused on how to find final KE

The first car, A alone, starts off with some velocity Vo. It collides with and 'sticks' to car B, and together they end up with some velocity Vo' that is a calculable fraction of Vo. The pair also retains the original momentum.

This pair then collides with car C, resulting in yet another velocity, say Vo''. It will be a fraction of the the velocity Vo'. Vo' and Vo'' can therefore be written in terms of the original Vo by substitution.

## 1. What causes cars to collide into each other?

There are several factors that can cause cars to collide into each other, such as driver error, mechanical failure, weather conditions, road hazards, and distractions.

## 2. How can collisions between cars be prevented?

Collisions between cars can be prevented by following traffic laws and regulations, maintaining a safe speed and distance while driving, avoiding distractions, regularly maintaining and inspecting your vehicle, and being aware of potential hazards on the road.

## 3. What are the most common types of collisions between cars?

The most common types of collisions between cars are rear-end collisions, side-impact collisions, and head-on collisions.

## 4. What are the potential injuries that can result from a car collision?

Injuries from car collisions can range from minor cuts and bruises to more serious injuries such as whiplash, broken bones, spinal injuries, and traumatic brain injuries.

## 5. How do scientists analyze car collisions?

Scientists analyze car collisions by examining physical evidence at the crash site, reconstructing the events leading up to the collision, and using computer simulations and models to understand the forces and impacts involved in the collision.

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