Question

If `e=(1+klamda)/(k(1-lambda))`, `0<lamda<1/2` and e is the coefficient of restitution, deduce that `k>1`?

I thought about putting this in Algebra since this is what it boils down to, but since its a collision problem and `e` is the coefficient of restitution, I thought it belongs in Physics more.

The original problem was:

A smooth sphere S of mass `m` is moving with speed `u` on a smooth horizontal plane. The sphere S collides with another smooth sphere T, of equal radius to S but of mass `km`, moving in the same straight line and in the same direction with speed `lamdau`, `0< lamda<1/2` . The coefficient of restitution between S and T is `e`.

Given that S is brought to rest by the impact, show that `e=(1+klamda)/(k(1-lambda))`. [I did this bit fine]

Deduce that `k>1`

Thank you!

Answer 1

See below

Explanation:

For this collision:

`e lt 1`

`implies (1 + k lambda)/(k(1 - lambda)) lt 1`

`k > 0` and `lambda < 1/2 implies 1 - lambda > 0`, ie the denominator is always positive, so we can make the next step whilst guaranteeing the same inequality sign:

`1 + k lambda lt k(1 - lambda)`

`implies 1 lt k(1 - 2 lambda) qquad square`

`lambda < 1/2 implies 1 - 2 lambda > 0`, so we can divide and guarantee the same inequality sign in `square`:

`implies k > 1/(1 - 2 lambda)`

Because `0 < lambda < 1/2` then:

• `0 <2 lambda < 1`

• `0 > - 2 lambda > -1`

• `1 >1 - 2 lambda > 0`

• `color(blue)(0 < 1 - 2 lambda < 1)`

So: `k = (1)/("a number between 0 and 1") implies k > 1`