To solve the problem regarding the three balls and their collisions, we need to apply the principles of momentum conservation and the coefficient of restitution. Let's break it down step by step to find the relationship between the masses of the balls.
Understanding the Collision Process
We have three balls with masses m1, m2, and m3 arranged in a straight line. The first ball (m1) moves towards the second ball (m2) and collides with it, coming to rest after the collision. The second ball then collides with the third ball (m3) and also comes to rest. The coefficient of restitution (e) is a measure of how elastic the collisions are, defined as the ratio of the relative velocity of separation to the relative velocity of approach.
First Collision: m1 and m2
When m1 strikes m2, it comes to rest. Let's denote the initial velocity of m1 as u1 and the initial velocity of m2 as u2 = 0 since it is at rest. After the collision, m1 stops, so its final velocity v1 = 0. The final velocity of m2 after the collision can be denoted as v2.
Using the conservation of momentum:
- Initial momentum = Final momentum
- m1 * u1 + m2 * u2 = m1 * v1 + m2 * v2
- m1 * u1 = m2 * v2
From this, we can express the final velocity of m2:
v2 = (m1/m2) * u1
Second Collision: m2 and m3
Now, m2 collides with m3. Before this collision, m2 has a velocity of v2 and m3 is at rest (u3 = 0). After the collision, m2 comes to rest (v2 = 0), and we denote the final velocity of m3 as v3.
Applying the conservation of momentum again:
- m2 * v2 + m3 * u3 = m2 * 0 + m3 * v3
- m2 * v2 = m3 * v3
Substituting for v2 from the first collision:
m2 * (m1/m2) * u1 = m3 * v3
From this, we can express the final velocity of m3:
v3 = (m1/m3) * u1
Relating the Masses
Now, we have two relationships:
- From the first collision: v2 = (m1/m2) * u1
- From the second collision: v3 = (m1/m3) * u1
Since both collisions involve the same initial velocity u1, we can set up a ratio:
m1/m2 = m2/m3
Rearranging gives us:
m1 * m3 = m2 * m2
Final Relationship
This leads us to conclude that:
m1/m2 = m2/m3
Thus, the ratio of the masses m1 to m2 is the same as the ratio of m2 to m3. Therefore, the correct answer to the question is:
Option a) m2/m3
In summary, the relationships derived from the conservation of momentum and the coefficient of restitution show that the mass ratios are consistent across the collisions, leading us to the conclusion that the ratio of m1 to m2 is equal to the ratio of m2 to m3.
