A ball with a mass of # 2 kg# is rolling at #25 m/s# and elastically collides with a resting ball with a mass of # 2 kg#. What are the post-collision velocities of the balls?

Question

A ball with a mass of # 2  kg# is rolling at #25   m/s# and elastically collides with a resting ball with a mass of #   2    kg#. What are the post-collision velocities of the balls?

Layla Ahmed · Accepted Answer

The velocities of the balls are #=0ms^-1# and #=25ms^-1#

As the collision is elastic, there is conservation of linear momentum and conservation of kinetic energy

Let the velocity before collision be #u_1ms^-1# and #u_2ms^-1#

Let the velocity after collision be #v_1ms^-1# and #v_2ms^-1#

#m_1u_1+m_2u_2=m_1v_1+m_2v_2#

#1/2m_1u_1^2+1/2m_2u_2^2=1/2m_1v_1^2+1/2m_2v_2^2#

Here, we have

#m_1=m_2=2kg#

#u_1=25ms^-1#

#u_2=0ms^-1#

Therefore,

#v_1+v_2=25#

#v_1^2+v_2^2=25^2=625#

#v_1=25-v_2#

#(25-v_2)^2+v_2^2=625#

#625-50v_2+v^2+v^2=625#

#2v_2^2-50v_2=0#

#2v_2(v_2-25)=0#

#v_2=25# and #v_1=0#

William Audy · Answer

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

First, calculate the total momentum before the collision. Then, use the conservation of momentum to find the total momentum after the collision.

Next, use the conservation of kinetic energy to find the post-collision velocities.

Let's denote:
- $m_1$ and $m_2$ as the masses of the two balls
- $v_1$ and $v_2$ as the velocities of the two balls before the collision
- $v_{1f}$ and $v_{2f}$ as the velocities of the two balls after the collision

Given:
- $m_1 = m_2 = 2 \, \text{kg}$
- $v_1 = 25 \, \text{m/s}$
- $v_2 = 0 \, \text{m/s}$ (since the second ball is at rest initially)

Using conservation of momentum: 
$$ m_1v_1 + m_2v_2 = m_1v_{1f} + m_2v_{2f} $$

Using conservation of kinetic energy: 
$$ \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 = \frac{1}{2} m_1 v_{1f}^2 + \frac{1}{2} m_2 v_{2f}^2 $$

Solve these two equations to find $v_{1f}$ and $v_{2f}$.

After solving, you should get:
- $v_{1f} \approx 0 \, \text{m/s}$
- $v_{2f} \approx 25 \, \text{m/s}$