Question

Is a head on collision between two cars more damaging to the occupants?

Answer 1

Hint: When two cars collide, the collision can be elastic or inelastic. In case of an elastic collision, the two cars recoil after the impact and as a consequence, both the cars experience a change in the direction of their velocities indicating a huge change in momentum, which leads to a larger impulsive force and hence greater damage. If an inelastic collision occurs, the cars stick together after impact, and their velocities become zero. Therefore, each car’s change in momentum is lesser, and lower damage is caused to the occupants.

Formula used::
Condition for conservation of energy:
\[\dfrac{1}{2}{m_1}u_1^2 + \dfrac{1}{2}{m_2}u_2^2 = \dfrac{1}{2}{m_1}v_1^2 + \dfrac{1}{2}{m_2}v_2^2\]
Condition for conservation of linear momentum:
\[{m_1}{u_1} + {m_2}{u_2} = {m_1}{v_1} + {m_2}{v_2}\]
Where \[{m_1},{u_1},{v_1}\] are the mass and initial and final velocities of the first object ( say car 1) and \[{m_2},{u_2},{v_2}\] are the mass and initial and final velocities of the second object ( say car 2).

Complete step by step solution:
Given: Collisions are broadly classified into two categories, elastic and inelastic collisions. When there is no net loss in kinetic energy after the collision, such type of collision is termed as elastic collision. If there is a loss of kinetic energy or the energy is changed into a different type of energy as a result of collision, then the collision is termed inelastic. Kinetic energy is conserved in elastic collision only. Momentum is conserved in both elastic and inelastic collisions.

The conditions for elastic collision are:
- The total kinetic energy before and after the collision remains unchanged, that is the kinetic energy of the system is conserved.
- The total linear momentum before and after the collision remains unchanged, that is linear momentum of the system is conserved.

Thus, using the condition for elastic collision we get,
From conservation of kinetic energy,
\[\dfrac{1}{2}{m_1}u_1^2 + \dfrac{1}{2}{m_2}u_2^2 = \dfrac{1}{2}{m_1}v_1^2 + \dfrac{1}{2}{m_2}v_2^2\]----- (1)
From the conservation of linear momentum,
\[{m_1}{u_1} + {m_2}{u_2} = {m_1}{v_1} + {m_2}{v_2}\]----- (2)
If elastic collision occurs, all kinetic energy is conserved and hence impact is greater and this situation is more alarming. If inelastic collision occurs, the total kinetic energy of the system is not conserved and hence the resultant velocities are either zero or reduced. Hence the impact is lesser and the situation is less dangerous.

Hence, less damage is caused to the occupants.

Note: This is not equivalent to a car of mass equal to sum of masses of car 1 and car 2 and initial velocity a sum of the initial velocities of the two cars. This is more similar to a car 1 of some velocity hitting a stationary car 2 with initial velocity of car 1 as given in the problem. Then also, the answer is the same.