Question

4. During an elastic collision between two balls, the force-time plot has Gaussian profile. Provide the equations of velocities of both balls during the collision.

Answer 1

use the law of conservation of momentum
\[m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2 \]

Answer 2

during the collision, force is a gaussian function of time (given).
let this function be f(t)
At any instant of time, for both balls,
\[m*dv/dt = f(t)\]

let initial velocities be u1, u2 and masses m2, m2. let final velocites be v1, v2.
then,
conservation of momentum,
\[m1*u1 + m2*u2 = m1*v1 + m2*v2\]

and conservation of energy says,
\[1/2m1*u1^2 + 1/2m2*u2^2 = 1/2m1*v1^2 + 1/2m2*v2^2\]

so we can calculate the final velocites v1, v2 from above.

What the question really asks is - What are the velocites during the collision.
\[dv/dt = f(t)/m1\]
integration from time = 0 to arbitrarty time (t) during the collsion.
limits are at t = 0, v = u1 , at time = t, velocity = v
\[\int\limits_{u1}^{v} dv = 1/m1*\int\limits_{0}^{t}f(t)dt\]

Similarly for mass m2,
\[\int\limits_{u2}^{v} dv = 1/m1*\int\limits_{0}^{t}f(t)dt\]

These are the equations of velocities DURING the collision

Answer 3

Thanks zaphodplaysitsafe

Answer 4

you're welcome, kindly note that by mistake, for the second mass, right at the last, where i wrote the equation for velocity of 2nd mass, i divided by m1. ofcourse, it should be m2 :)