To solve this problem, we need to analyze the situation using the principles of conservation of momentum and energy. We have three equal masses, m1, m2, and m3, where m1 collides with m2 and sticks to it, creating a system that then interacts with m3. Here's how we can break this down step by step.
Understanding the Initial Conditions
In the beginning, we have:
- Masses m2 and m3 are stationary.
- Mass m1 is moving with an initial velocity v.
- The spring is uncompressed, meaning it has no potential energy at this moment.
Applying Conservation of Momentum
When mass m1 collides with mass m2, which is initially at rest, they stick together. Therefore, we can apply the principle of conservation of momentum to this collision. The total momentum before the collision must equal the total momentum after the collision.
The momentum before the collision is:
P_initial = m1 * v + m2 * 0 = m1 * v
After the collision, the two masses m1 and m2 move together with a common velocity (let's denote it as V). So, the momentum after the collision is:
P_final = (m1 + m2) * V
Setting the initial and final momenta equal gives us:
m1 * v = (m1 + m2) * V
Finding the Common Velocity
Since m1 = m2 = m, we can simplify our equation:
m * v = (m + m) * V
m * v = 2m * V
Now, we can solve for V:
V = v / 2
Energy Transfer to Mass m3
Now that we know the velocity of the combined mass m1 and m2 after the collision, we need to consider what happens to mass m3. As mass m1 and m2 move towards mass m3, they compress the spring, converting kinetic energy into potential energy in the spring.
Using Conservation of Energy
The kinetic energy of the combined mass m1 and m2 before they hit m3 can be expressed as:
KE_initial = (1/2) * (m1 + m2) * V^2 = (1/2) * (2m) * (v/2)^2 = (1/2) * 2m * (v^2 / 4) = (mv^2) / 4
When the spring is maximally compressed, all this kinetic energy is converted to potential energy stored in the spring:
PE_spring = (1/2) * k * x^2
At the moment of maximum compression, we can relate the maximum velocity of m3 (let's denote it as V_m3) to the potential energy in the spring. When the spring releases its energy, this will push mass m3 forward. Thus, the kinetic energy of m3 when the spring is released is:
KE_m3 = (1/2) * m3 * V_m3^2 = (1/2) * m * V_m3^2
Equating Energies
At maximum compression, we have:
KE_initial = KE_m3
(mv^2) / 4 = (1/2) * m * V_m3^2
We can cancel out m from both sides, yielding:
v^2 / 4 = (1/2) * V_m3^2
Multiplying both sides by 2 gives:
v^2 / 2 = V_m3^2
Solving for the Maximum Velocity of m3
Now, taking the square root of both sides, we find:
V_m3 = v / √2
This result shows that the maximum velocity of mass m3, after the spring has fully expanded and released its energy, is v divided by the square root of 2. This makes sense as the energy is being shared among the three masses.
Summary of Results
In summary, when mass m1 collides with mass m2 and they stick together, they create a combined mass that moves with half the original velocity of m1. When they interact with mass m3 and the spring, the energy is transferred, and we find that the maximum velocity of mass m3 is given by:
V_m3 = v / √2
This equation illustrates how the conservation of momentum and energy principles work together in this collision and spring system. If you have any further questions about any of the steps or concepts, feel free to ask!
