In a head-on elastic collision of two bodies of equal massesa)the velocities are interchangedb)the speeds are interchangedc)the momenta are interchangedd)the faster body slows down and the slower body speeds upCorrect answer is option 'A,B,C,D'. Can you explain this answer?

Question

Answer

As momentum is conserved,

and masses are equal.

as collision is perfectly elastic. Therefore,

coefficient of restitution, e = 1 ⇒ v_approach = v_separation

By solving this,

Hence, the velocities are interchange including their magnitude(means speeds),

u₁ = v₂ and v₁ = u₂.

Multiply velocities with mass,

Hence, momentums are also interchanged.

Since speeds are exchanged. Hence, the faster body slows down and the slower body speeds up.

Answer

As momentum is conserved,

and masses are equal.

as collision is perfectly elastic. Therefore,

coefficient of restitution, e = 1 ⇒ v_approach = v_separation

By solving this,

Hence, the velocities are interchange including their magnitude(means speeds),

u₁ = v₂ and v₁ = u₂.

Multiply velocities with mass,

Hence, momentums are also interchanged.

Since speeds are exchanged. Hence, the faster body slows down and the slower body speeds up.

Answer

Understanding Head-On Elastic Collisions
In a head-on elastic collision involving two bodies of equal masses, several important principles apply. Here’s a breakdown of the outcomes:
1. Velocities are Interchanged
- In an elastic collision, the law of conservation of momentum and kinetic energy applies.
- When two bodies of equal mass collide, their velocities effectively swap.
- If Body A moves towards Body B with velocity V1 and Body B moves towards Body A with velocity V2, after the collision, Body A will take on the velocity of Body B and vice versa.
2. Speeds are Interchanged
- Since the masses are equal, the speeds (magnitude of velocity) are also exchanged.
- This means if Body A was faster, it slows down while Body B speeds up to match the speed of Body A prior to the collision.
3. Momenta are Interchanged
- The momentum of a body is the product of its mass and velocity.
- In this case, because the masses are equal and velocities are interchanged, the momenta are also effectively swapped between the two bodies.
4. The Faster Body Slows Down and the Slower Body Speeds Up
- The kinetic energy, which is proportional to the square of the speed, is conserved in elastic collisions.
- As a result, the faster body loses some speed while the slower body gains speed, leading to an overall exchange of kinetic energy.
In summary, for two equal-mass bodies in a head-on elastic collision, the outcomes are interconnected: velocities, speeds, and momenta are interchanged, and each body adjusts its speed accordingly based on its initial conditions.

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