Question

An iron ball a of radius r collides head on with another stationary iron ball b of radius 2r. Given, the coefficient of restitution e

Answer 1

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Answer 2

The magnitude of the ratio of their speeds just after the collision is 2.

Explanation:

Given that,

Radius of ball a = r

Radius of ball b = 2r

Suppose we determine the magnitude of the ratio of their speeds just after the collision

If the coefficient of restitution is 0.5.

Let the density is constant.

So, $mass\propto r^3$

$m_{1}=\dfrac{4}{3}\pi r^3\times \rho=m$

$m_{2}=\dfrac{4}{3}\pi 8r^3\times \rho= 8m$

Using conservation of momentum

$m_{i}u_{i}+m_{2}u_{i}=m_{1}v_{1}+m_{2}v_{2}$

$mv=mv_{1}+8mv_{2}$

$v=v_{1}+8v_{2}$....(I)

Using formula of restitution

$e=\dfrac{v_{2}-v_{1}}{v}$

Put the value into the formula

$0.5v=v_{2}-v_{1}$...(II)

From equation (I) and (II)

$1.5v=9v_{2}$

$v_{2}=\dfrac{1.5v}9}$

Put the value of v₁ in the equation (I)

$v_{1}=\dfrac{1.5v}{9}-0.5v$

$v_{1}=\dfrac{-3v}{9}$

We need to calculate the magnitude of the ratio of their speeds just after the collision

$\dfrac{v_{1}}{v_{2}}=\dfrac{\dfrac{-3v}{9}}{\dfrac{1.5v}{9}}$

$\dfrac{v_{1}}{v_{2}}=\dfrac{-3}{1.5}$

$\dfrac{v_{1}}{v_{2}}=2$

Hence, The magnitude of the ratio of their speeds just after the collision is 2.

Learn more :

Topic : restitution

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