Answer:
![\frac{inertia_B}{inertia_A}=9](https://tex.z-dn.net/?f=%5Cfrac%7Binertia_B%7D%7Binertia_A%7D%3D9)
Explanation:
First of all, let's remind that:
- The kinetic energy of an object is given by
, where m is the mass and v is the speed
- The momentum of an object is given by ![p=mv](https://tex.z-dn.net/?f=p%3Dmv)
- The inertia of an object is proportional to its mass, so we can write
, where k just indicates a constant of proportionality
In this problem, we have:
-
(the two objects have same kinetic energy)
-
(A has three times the momentum of B)
Re-writing both equation we have:
![\frac{1}{2}m_A v_A^2 = \frac{1}{2}m_B v_B^2\\m_A v_A = 3 m_B v_B](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7Dm_A%20v_A%5E2%20%3D%20%5Cfrac%7B1%7D%7B2%7Dm_B%20v_B%5E2%5C%5Cm_A%20v_A%20%3D%203%20m_B%20v_B)
If we divide first equation by second one we get
![v_A = 3 v_B](https://tex.z-dn.net/?f=v_A%20%3D%203%20v_B)
And if we substitute it into the first equation we get
![m_A (3 v_B)^2 = m_B v_B^2\\9 m_A v_B^2 = m_B v_B^2\\m_B = 9 m_A](https://tex.z-dn.net/?f=m_A%20%283%20v_B%29%5E2%20%3D%20m_B%20v_B%5E2%5C%5C9%20m_A%20v_B%5E2%20%3D%20m_B%20v_B%5E2%5C%5Cm_B%20%3D%209%20m_A)
So, B has 9 times more mass than A, and so B has 9 times more inertia than A, and their ratio is:
![\frac{I_B}{I_A}=\frac{km_B}{km_A}=\frac{9m_A}{m_A}=9](https://tex.z-dn.net/?f=%5Cfrac%7BI_B%7D%7BI_A%7D%3D%5Cfrac%7Bkm_B%7D%7Bkm_A%7D%3D%5Cfrac%7B9m_A%7D%7Bm_A%7D%3D9)