A ball with a mass of # 5 kg# is rolling at #6 m/s# and elastically collides with a resting ball with a mass of #9 kg#. What are the post-collision velocities of the balls?

Question

A ball with a mass of # 5    kg# is rolling at #6   m/s# and elastically collides with a resting ball with a mass of #9  kg#. What are the post-collision velocities of the balls?

Julia Aguilar · Accepted Answer

#v_1=-12/7 m/s#
#v_2=30/7m/s# #5*6+0=5*v_1+9*v_2" Conversation of momentum"# #30=5*v_1+9*v_2 " (1)"# #6+v_1=0+v_2# #"'the sum of velocities of objects before and after collision is equal'" # #v_2=6+v_1" (2)"# #30=5*v_1+9*(6+v_1)# #30=5*v_1+54+9*v_1# #30-54=14*v_1# #-24=14*v_1# #v_1=-24/14# #v_1=-12/7 m/s# #"use (2)"# #v_2=6-12/7# #v_2=30/7m/s#

Carson Alexander · Answer

undefined To find the post-collision velocities of the balls, we can use the conservation of momentum and the conservation of kinetic energy.

Let $v_1$ and $v_2$ be the velocities of the 5 kg and 9 kg balls, respectively, after the collision.

According to the conservation of momentum:

$$ m_1v_{1i} + m_2v_{2i} = m_1v_{1f} + m_2v_{2f} $$

And since it's an elastic collision, we also have:

$$ \frac{1}{2}m_1v_{1i}^2 + \frac{1}{2}m_2v_{2i}^2 = \frac{1}{2}m_1v_{1f}^2 + \frac{1}{2}m_2v_{2f}^2 $$

Given:
$ m_1 = 5 $ kg (mass of the rolling ball),
$ m_2 = 9 $ kg (mass of the resting ball),
$ v_{1i} = 6 $ m/s (initial velocity of the rolling ball), and
$ v_{2i} = 0 $ m/s (initial velocity of the resting ball).

Plugging in the values and solving the equations, we can find the post-collision velocities:

$$ v_{1f} = \frac{(m_1 - m_2)v_{1i} + 2m_2v_{2i}}{m_1 + m_2} $$
$$ v_{2f} = \frac{(m_2 - m_1)v_{2i} + 2m_1v_{1i}}{m_1 + m_2} $$

Substituting the given values:
$$ v_{1f} = \frac{(5 - 9) \times 6 + 2 \times 9 \times 0}{5 + 9} $$
$$ v_{2f} = \frac{(9 - 5) \times 0 + 2 \times 5 \times 6}{5 + 9} $$

After calculation:
$$ v_{1f} = \frac{-24}{14} \approx -1.71 \, \text{m/s} $$
$$ v_{2f} = \frac{60}{14} \approx 4.29 \, \text{m/s} $$

So, the post-collision velocities of the balls are approximately $ -1.71 \, \text{m/s} $ and $ 4.29 \, \text{m/s} $, respectively.