A ball with a mass of #2 kg# is rolling at #6 m/s# and elastically collides with a resting ball with a mass of #1 kg#. What are the post-collision velocities of the balls?

Question

A ball with a mass of #2  kg# is rolling at #6  m/s# and elastically collides with a resting ball with a mass of #1 kg#. What are the post-collision velocities of the balls?

William Audy · Accepted Answer

#v_1^'=2 m/s#
#v_2^'=8 m/s# #vec P_1 + vec P_2= vec P_2^' + vec P_2^'" the conservation of momentum"# #m_1*v_1+m_2*v_2=m_1*v_1^'+m_2*v_2^'# #2*6+0=2*v_1^'+1*v_2^'# #12=2*v_1^'+v_2^'" (1)"# #v_1+v_1^'=v_2+v_2^'" (2)# #6+v_1^'=0+v_2^'# #v_2^'=6+v_1^'" (3)"# #"use equation (1)"# #12=2*v_1^'+6+v_1^'# #12-6=3*v_1^2# #6=3*v_1^' " " v_1^'=2 m/s# #"use equation (3)"# #v_2^'=6+2# #v_2^'=8 m/s#

Charlotte Aaron · Answer

undefined To solve this problem, we can use the principle of conservation of momentum and the principle of conservation of kinetic energy.

Let $ m_1 $ be the mass of the first ball (2 kg), $ v_{1i} $ be the initial velocity of the first ball (6 m/s), $ m_2 $ be the mass of the second ball (1 kg), and $ v_{2i} $ be the initial velocity of the second ball (0 m/s).

Using the conservation of momentum:

$$ m_1 \cdot v_{1i} + m_2 \cdot v_{2i} = m_1 \cdot v_{1f} + m_2 \cdot v_{2f} $$

$$ 2 \cdot 6 + 1 \cdot 0 = 2 \cdot v_{1f} + 1 \cdot v_{2f} $$

$$ 12 = 2 \cdot v_{1f} + v_{2f} $$

Using the conservation of kinetic energy:

$$ \frac{1}{2} m_1 \cdot (v_{1i})^2 + \frac{1}{2} m_2 \cdot (v_{2i})^2 = \frac{1}{2} m_1 \cdot (v_{1f})^2 + \frac{1}{2} m_2 \cdot (v_{2f})^2 $$

$$ \frac{1}{2} \cdot 2 \cdot (6)^2 + \frac{1}{2} \cdot 1 \cdot (0)^2 = \frac{1}{2} \cdot 2 \cdot (v_{1f})^2 + \frac{1}{2} \cdot 1 \cdot (v_{2f})^2 $$

$$ 36 + 0 = 2 \cdot (v_{1f})^2 + \frac{1}{2} \cdot (v_{2f})^2 $$

$$ 36 = 2 \cdot (v_{1f})^2 + \frac{1}{2} \cdot (v_{2f})^2 $$

Now, we have a system of equations:

$$ 12 = 2 \cdot v_{1f} + v_{2f} $$
$$ 36 = 2 \cdot (v_{1f})^2 + \frac{1}{2} \cdot (v_{2f})^2 $$

Solving this system of equations will give us the post-collision velocities of the balls.