If two bodies of equal masses undergo elastic collisions in one dimension , then after the collision the body will exchange their velocities?

Question

Answer

Explanation:

When two bodies of equal masses undergo elastic collisions in one dimension, the following happens:

The bodies approach each other with equal and opposite velocities.

At the point of collision, the bodies exert equal and opposite forces on each other.

These forces cause the bodies to accelerate towards each other, and then away from each other.

During the collision, the total momentum of the system is conserved.

After the collision, the bodies will exchange velocities if they are of equal masses and the collision is perfectly elastic.

Proof:

Let us consider two bodies A and B of equal masses m1 and m2 respectively. Let us assume that A is moving towards B with velocity u1 and B is moving towards A with velocity u2. After the collision, A moves away from B with velocity v1 and B moves away from A with velocity v2.

According to the law of conservation of momentum:

m1u1 + m2u2 = m1v1 + m2v2

Since the masses of A and B are equal, we can simplify this equation to:

u1 + u2 = v1 + v2

Now, according to the law of conservation of kinetic energy:

(1/2)m1u1^2 + (1/2)m2u2^2 = (1/2)m1v1^2 + (1/2)m2v2^2

Since the masses of A and B are equal, we can simplify this equation to:

u1^2 + u2^2 = v1^2 + v2^2

Now, let us add the two equations:

u1 + u2 + u1^2 + u2^2 = v1 + v2 + v1^2 + v2^2

Since the collision is elastic, we know that the kinetic energy of the system is conserved. This means that u1^2 + u2^2 = v1^2 + v2^2. Substituting this in the above equation, we get:

u1 + u2 = v1 + v2

This means that after the collision, the bodies will exchange velocities if they are of equal masses and the collision is perfectly elastic.

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