Question

Elastic collision in one dimension 1 derivation

Answer 1

The Problem:

A particle of mass m1 and velocity v collides elastically (in one dimension) with a stationary particle of mass m2. What are the velocities of m1 and m2 after the collision?

Before diagram  in attachment

A particle of mass m1 and velocity v collides elastically with a particle of mass m2, initially at rest.

After the collision, m1 has velocity v1, and m2 has velocity v2. What are v1 and v2?

The Solution:

Since this is an isolated system, the total momentum of the two particles is conserved:

equation 1

Also, since this is an elastic collision, the total kinetic energy of the 2-particle system is conserved:

conservation of KE equation

Multiplying both sides of this equation by 2 gives:

Conservation of KE equation

Suppose we solve equation 1 for v2:

equation 3

and then substitute this result into equation 2:

equation 2 expanded

Expanding and multiplying both sides by m2 in order to clear fractions gives:

equation 3b

Now, gather up like terms of v1:

Equation 4

Notice that equation 4 is a standard quadratic in v1, like Ax2 + Bx + C = 0, where:

quadratic coefficients

So, we can use the quadratic formula (quadratic formula) to solve for v1:

v1 = quadratic formula

Inside the radical, the last term of the discriminant has factors like (a + b)(a - b) = a2 - b2, so:

equation 4c

Now, expand and simplify:

Equation 5

So, there are 2 solutions (of course...). Taking the positive sign in the numerator of equation 5 gives:

v1 = v

Physically, this means that no collision took place - the velocity of m1 was unchanged. That isn't the solution we have come this far to find. Taking the negative sign in the numerator of equation 5 gives:

equation 6

That's it! Now, to find v2, substitute equation 6 into equation 3:

solution for v2

There it is! Equations 6 and 7 give the velocities of the two particles after the collision.

solution summary

Answer 2

Answer:

Elastic collision in One dimension:-)

Consider two bodies having masses M1 and M2 moving along a straight line such that line joining their centres is parallel or anti parallel to the velocity of the bodies before and after the collision.

ln elastic collision both Momentum as well as kinetic energy are conserved. by using this we can form two equations and solve them to find values of velocities after collision as :-